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SYNOPSIS 


OF 


LINEAR  ASSOCIATIVE  ALGEBRA 


A  RHPORT  ON  ITS  NATURAL  DEVELOPMENT  AND   RESULTS  REACHED 

UP  TO  THE   PRESENT  TIME 


BY 


JAMES  BYRNIE  SHAW 

Professor  of  Mathematics  in  the  James  MiUikin  University 


A 


WASHINGTON,  D.  C. : 

Published  by  the  Carnegie  Institution  of  Washington 
1907 


CARNEGIE  INSTITUTION  OF  WASHINGTON 

Publication  No.  78 


Z^t  Boti  (gattimcxe  {prtee 

BALTIMORE,  MD. ,  U.  B.  A. 


CONTENTS. 


iKTIiODUCrilON. 


PAOB 

5 


Part  I.     General  Theory. 

I.      Deliiiitioiis.  ...... 

1 1.     The  cliaracteristic  equations  of  a  number. 
I II.     'I'lic  characteristic  equations  of  the  algebra. 
Associative  units.  .         .         .         .         . 

Sub-algebras,  Redncibiiity,  Deletion. 
Dedekind  and  Fkobenius  algebras. 


IV. 

\^ 

VI. 

VII. 

VIII. 

IX. 

X. 

xr. 

XII. 


Scheffers  and  Peiuce  algebras. 
Kkoneckeu  and  WEiEKf5TRASS  algebras. 
Algebras  with  coefficients  in  arbitrary  fields. 

Dickson  algebras. 
Number  theory  of  algebras. 
Function  theory  of  algebras. 
Cirou])  theory  of  algebras. 


Kea 


algebri 


IS. 


XIII.     General  theory  of  algebras. 

Part  II.    Particular  Algebras. 

XIV^.     Complex  numbers 

Quaternions.  ..... 

Alternate  algebras.       .... 

Biquaternions  or  octonions. 
Triquaternions  and  (^uadriquaternions. 
Sylvester  algebras. 
Peirce  algebras.  .... 

ScHEFFERS  algebras. 

Caht.\n  algebras.  .... 


XV. 

XVI. 

XVII. 

XVIII. 

XIX. 

XX. 

XXI. 

XXII. 


Part  III.     Applications. 

XXIII.  Geometrical  applications. 

XXIV.  Physico-mechanical  applications 
XXV.     Transformation  groups. 

XXVI.     Abstract  groups. 
XXVII.     (Special  classes  of  groups. 
XXVIII.     Differential  equations. 
XXIX.     Modular  systems. 
XXX.     Operators. 
Bibliography 


9 
31 
.3.5 
40 
44 
48 
52 
56 

58 
60 
68 

72 


79 

80 

83 

87 

91 

93 

101 

107 

HI 


113 
120 
120 
125 
129 
1.33 
133 
134 
1.35 


ERRATA. 

Page. 

11.  Line  13,  for  \e,j\'  read   \(\j\\ 

15.  In  the  foot-uotes  change  numbering  as  follows:  for  1  read  2,  for  2  read  3,  for  3  read  4, 

for  4  read  1. 

26.  Line  21,  for  A"  read  h^  . 

33.  Line  15,  for  AejC^  read  ^e,f.. 

34.  Line  6,  for  [?«,  (,",)  read  [?«,'(;",). 
49.  Line  6,  for  ?h,"|i  read  ?w|  +  ,. 

53,  54.     In  the  table  for  r  >6  in  every  instance  change  r—2  to  r—3,  and  r— 3  to  r  — 4. 

In  case  (27),  hoioever,  read  e,  =  (311)  —  (12  r  —  3). 
57.     Line  8,  for  t,  read  t^ . 
59.     Line  33,  remove  the  period  after  A. 

67.  Line  12,  insert  a  comma  (,)  after  "integer". 

68.  Lines  9  and  10,  cliange  y  to  ■-•. 

71.  Line  17,  in  type  III  for  e,,  read  e^. 

72.  Last  line,  for  a  q  «~'  read  a  q  «"'. 

73.  Line  3  from  bottom,  for  jk  <■    read  jk '  . 
94.     Line  7,  for  Srj'  read  SPj"'. 

94.     Last  line,  in  the  second  column  of  the  determinant  and  third  line  for  S.  /"'  a  j^  <p  ~<p 

read  S  .j''  nj''  <p  . 
100.     Line  12,  for  (t>  =  <p  read  </>  =  0- 

106.  Some  of  these  cases  are  equivalent  to  others  previously  given. 

107.  Line  3  from  bottom,  /or  e.^  =  (221)  read  e^  =  (211). 

116.     Line  25  should  read  p  =  '- ^r~^ —  . 

124.     Note  3,  add:  of.  Beez  '-'. 

128.     Line  11,  for  i  =  I k^  read  i  =  1 ....//, . 


SYNOPSIS  OF  LINEAR  ASSOCIATIVE  ALGEBRA 


INTRODUCTION. 


This  memoir  is  genetic  in  its  intent,  in  that  it  aims  to  set  forth  the  present 
state  of  the  mathematical  discipline  indicated  by  its  title:  not  in  a  comparative 
study  of  different  known  algebras,  nor  in  the  exhaustive  study  of  any  particular 
algebra,  but  in  tracing  the  general  laws  of  the  whole  subject.  Developments 
of  individual  known  algebras  may  be  found  in  the  original  memoirs.  A  partial 
bibliography  of  this  entire  field  may  be  found  in  the  Bibliography  of  the 
Quaternion  Society,^  which  is  fairly  complete  on  the  subject.  Comparative 
studies,  more  or  less  complete,  may  be  found  in  Hankel's  lectures,^  and  in 
Cayley's  paper  on  Multiple  Algebra.*  These  studies,  as  well  as  those  men- 
tioned below,  are  historical  and  critical,  as  well  as  comparative.  The  phyletic 
development  is  given  partially  in  Study's  Encyklopildie^  article,  his  CJiicufjo 
Congress'''  paper,  and  in  Cartan's  Encyclopedic^  article.  These  papers  furnish 
numerous  expositions  of  systems,  and  references  to  original  sources.  Further 
historical  references  are  also  indicated  below. ** 

In  view  of  this  careful  work  therefore,  it  does  not  seem  desirable  to  review 
the  field  again  historically.  There  is  a  necessity,  however,  for  a  presentation 
of  the  subject  which  sets  forth  the  results  already  at  hand,  in  a  genetic  order. 
From  such  presentation  may  possibly  come  suggestions  for  the  future. 
Attention  will  be  given  to  chronology,  and  it  is  hoped  the  references  given 
will  indicate  prioritv  claims  to  a  certain  extent.  These  are  not  always  easy 
to  settle,  as  they  are  sometimes  buried  in  papers  never  widely  circulated,  nor 
is  it  always  possible  to  say  whether  a  notion  existed  in  a  paper  explicitly  or 
only  implicitly,  consequently  this  memoir  does  not  presume  to  offer  any  authori- 
tative statements  as  to  priority. 

The  memoir  is  divided  into  three  parts :  General  Theory,  Particular  Si/s- 
temSy  Applications.  Under  the  General  Theory  is  given  the  development  of  the 
subject  from  fundamental  principles,  no  use  being  made  of  other  mathematical 
disciplines,  such  as  bilinear  forms,  matrices,  continuous  groups,  and  the  like. 

'Presented,  in  a  slightly  diflerent  form,  as  an  abstract  of  this  paper,  to  the  Congress  of  Arts  and 
Sciences  at  the  Universal  Exposition,  St.  Louis,  Sept.  33,  1904. 

'Bibliography  of  Quaternions  and  allied  systems  of  mathematics,  Alexander  Macfarlane,  1904,  Dublin. 

'Hankbl  1.  References  to  the  bibliography  at  the  end  of  the  memoir  are  given  by  author  and 
number  of  paper. 

*Caylet9.  'StudiS.  6Stcdt7.  'CabtasS. 

•Beman  3,  GiBBS  2,  R.  Graves  1,  Haqen  1,  Macfablane  4. 


6  SYNOPSIS  OF  LINEAR  ASSOCIATIVE  ALGEBRA 

We  find  the  first  such  general  treatment  in  Hamilton's  theory^  of  sets.  The 
first  extensive  attempt  at  development  of  algebras  in  this  way  was  made  by 
Benjamin  Pkirce^.  His  memoir  was  really  epoch-making.  It  lias  been  critic- 
ally examined  by  Hawkes^,  who  has  nndertaken  to  extend  Peirce's  method, 
showing  its  full  power*.  The  next  treatment  of  a  similar  character  was  by 
Cartan',  who  used  the  characteristic  equation  to  develop  several  theorems  of 
much  generality.  In  this  development  appear  the  scmi-simjjle,  or  Dedekind, 
and  the  jJseiwZo-nu?,  or  nilpotent,  sub-algebras.  The  very  important  theorem 
that  the  structure  of  every  algebra  may  be  represented  by  the  use  of  double 
units,  the  first  factor  being  quadrate,  the  second  non-quadrate,  is  the  ultimate 
proposition  he  reaches.  The  latest  direct  treatment  is  by  Taber",  who 
reexamines  the  results  of  Peirce,  establishing  them  fully  (which  Peirce  had 
not  done  in  every  case)  and  extending  them  to  any  domain  for  the  coordinates. 
[His  units  however  are  linearly  independent  not  only  in  the  field  of  the 
coordinates,  but  for  anj'  domain  or  field  ] 

Two  lines  of  development  of  linear  associative  algebra  have  been  followed 
besides  this  direct  line.  The  first  is  by  use  of  the  continuous  group.  It  was 
PoiNCARE^  who  first  announced  this  isomorphism.  The  method  was  followed 
by  ScHEFFERS*,  who  classified  algebras  as  quaternionic  and  noii  quaternionic. 
In  the  latter  class  he  found  "regular"  units  which  can  be  so  arranged  that 
the  product  of  any  two  is  exj^ressible  linearly  in  terms  of  those  which 
follow  both.  He  worked  out  complete  lists  of  all  algebras  to  order  five 
inclusive.  His  successor  was  Molien®,  who  added  the  theorems  that  quater- 
nionic algebras  contain  independent  quadrates,  and  that  quaternionic  algebras 
can  be  classified  according  to  non-quaternionic  types.  He  did  not,  however, 
reach  the  duplex  character  of  the  units  found  by  Cartan. 

The  other  line  of  development  is  by  using  the  matrix  theory.  C.  S.  Peirce^" 
first  noticed  this  isomorphism,  although  in  embryo  it  appeared  sooner.  The 
line  was  followed  by  Shaw  "  and  Frobenius  ^'\  The  former  shows  that  the 
equation  of  an  algebra  determines  its  quadrate  units,  and  certain  of  the  direct 
units;  that  the  other  units  form  a  nilpotent  system  which  with  the  quadrates 
may  be  reduced  to  certain  canonical  forms.  The  algebra  is  thus  made  a  sub- 
algebra  under  the  algebra  of  the  associative  units  used  in  these  canonical  forms. 
Frobenius  proves  that  every  algebra  has  a  Dedekind  sub-algebra,  whose 
equation  contains  all  factors  in  the  equation  of  the  algebra.  This  is  the  semi- 
simple  algebra  of  Cartan.  He  also  showed  that  the  remaining  units  form  a 
nilpotent  algebra  whose  units  may  be  regularized. 

It  is  interesting  to  note  the  substantial  identity  of  these  developments, 
aside  frojn  the  vehicle  of  expression.  The  results  will  be  given  in  the  order 
of  development  of  the  paper  with  no  regard  to  the  method  of  derivation.  The 
references  will  cover  the  difi'erent  proofs. 

'Hamilton  1.  'B.  Peirce  1,  3.  2IIawkes2.  *IIa\vkes  1,  8,  4. 

'  CaKTAN  2.  'TaDER  4.  '  POINCAHE   1.  'SCIIEFFEU3    1,   2,  8. 

•Moi.iENl.  '» C.  S.  Peirce  1,  4.  "  Shaw  4.  "Frodbnios  14. 


INTRODUCTION  7 

The  last  cluipter  of  tlic  general  llioory  gives  a  sketoli  of  the  theory  of 
general  algebra,  placing  linear  associative  algebra  in  its  genetic  relations  to 
general  linear  algebra.  Sonic  scant  work  has  been  done  in  this  development, 
particularly  along  the  line  of  symbolic  logic'  On  the  philosophical  side, 
which  this  general  treatment  leads  up  to,  there  have  always  been  two  views 
of  complex  algebra.  The  one  regards  a  number  in  such  an  algebra  as  in 
reality  a  duplex,  triplex,  or  multiplex  of  arithmetical  numbers  or  expressions. 
Tiie  so-called  units  become  mere  nmhrae  serving  to  distinguish  the  dilTerent 
coordinates.  This  seems  to  have  been  Cayley's^  view.  It  is  in  essence  the 
view  of  most  writers  on  the  subject.  The  other  regards  the  number  in  a  linear 
algebra  as  a  single  entity,  and  multiplex  only  in  that  an  equality  between 
two  such  numbers  implies  n  equalities  between  certain  coordinates  or  functions 
of  the  numbers.  This  was  Hamilton's''  view,  and  to  a  certain  extent  Gkass- 
mann's.'  The  first  view  seeks  to  derive  all  properties  from  a  multiplication 
table.  The  second  seeks  to  derive  these  properties  from  definitions  applying 
to  all  numbers  of  an  algebra.  The  attempt  to  base  all  mathematics  on  arith- 
metic leads  to  the  first  view.  The  attempt  to  base  all  mathematics  on  algebra, 
or  the  theory  of  entities  defined  by  relational  identities,  leads  to  the  second 
view.  It  would  seem  that  the  latter  would  be  the  more  profitable  from  the 
standpoint  of  utility.  This  has  been  the  case  notably  in  all  developments 
along  this  line,  for  example,  quaternions  and  space-analysis  in  general. 
Hamilton,  and  those  who  have  caught  his  idea  since,  have  endeavored  to  form 
expressions  for  other  algebras  which  will  serve  the  purpose  which  the  scalar, 
vector,  conjugate,  etc.,  do  in  quaternions,  in  relieving  the  system  of  reference 
to  any  unit-system.  Such  definition  of  algebra,  or  of  an  algebra,  is  a  develop- 
ment in  terms  of  what  may  be  called  the  fundamental  invariant  forms  of  the 
algebra.  The  characteristic  equation  of  the  algebra  and  its  derived  equations 
are  of  this  character,  since  they  are  true  for  all  numbers  irrespective  of  the 
units  which  define  the  algebra;  or,  in  other  words,  these  relations  are  identically 
the  same  for  all  equivalent  algebras.  The  present  memoir  undertakes  to  add  to 
the  development  of  this  view  of  the  subject. 

In  conclusion  it  may  be  remarked  that  several  theorems  occur  in  the  course 
of  the  memoir  which  it  is  believed  have  never  before  been  explicitly  stated. 
Where  not  perfectly  obvious  the  proof  is  given.  The  proofs  of  the  known 
theorems  are  all  indicated  by  the  references  given,  the  papers  referred  to  con- 
taining the  proofs  in  question.  No  fuller  treatment  could  properly  be  given 
in  a  synopsis. 

'  C.  S.  PeIRCE    I,  2,  SCIIUOEDER  1,    WHITEHEAD  1,  RUSSEI.L  1,  SHAW  t. 

«  Caylet  1,  9.     See  also  Gibus  1,  3,  3.  3 Hamilton  1,  3.  ■'Grassmaks  1,  2. 


PART  I.    GENERAL  THEORY. 
I.    DEFINITIONS. 

1.     EARLY  DEFINITIONS.' 

1.  Definitions.  Let  there  be  a  set  of  r  entities,  e,  .  .  .  .  e,,  which  will  be 
called  qualitative  units.  These  entities  will  serve  to  distinguish  certain  other 
entities,  called  coorc^wjafes,  from  each  other,  the  coordinates  belonging  to  a  given 
range,  or  ensemble  of  elements;  thus  if  a;  is  a  coordinate,  then  Mj^j  is  dilTerent 
from  OiCj,  if  i  ^J,  and  no  process  of  combination  belonging  to  the  range  of  «( 
can  produce  a^Cj  from  ajCj.  Thus,  the  range  may  be  the  domain  of  scalars 
(ordinary,  real,  and  imaginary  numbers),  or  it  may  be  the  range  of  integers,  or 
it  may  be  any  abstract  field,  or  even  any  algebra.  If  it  be  the  range  of  integers, 
subject  to  addition,  subtraction,  multiplication,  and  partially  to  division, 
then  by  no  process  of  this  kind  or  any  combination  of  such  can  ajCj  become 
ttiCj.  These  qnalijled  coordinates  may  be  combined  into  expressions  called 
complex,  or  hypercomplex,  or  multiple  numbers,  thus 

r 

a  ■=■  'S.  a^  gj 
t  =  i 

\i\  this  number  each  «,  is  supposed  to  run  through  the  entire  range.   The  units 

Cj,  or  le,,  are  said  to  define  a  region  of  order  r. 

2.  Theorems :  ^ 

(1)  (r(-|-i)ei  =  aei  + 6<?(,  and  conversely,  if  +  is  defined  for  the  range. 

(2)  Ofij  =  e,  0  =  0 ,     if  0  belongs  to  the  range. 

r 

(3)  1    aiei  =  0,     implies     0^  =  0  (i  =  1 r) 

1  =  1 

(4)  If  2   Oiff  =   S   6i<°i;  then  ai  =  6,.,  t  =  1 r,  and  conversely. 

1=1  1=1 

Theorems  (3)  and  (4)  might  be  omitted  by  changing  the  original  definitions, 
in  which  case  relations  might  exist  between  the  units.  Thus,  the  units  +  1 
and  —  1  are  connected  by  the  relation  +  1  +  ( —  1)  =  0. 

Algebras  of  this  character  have  more  units  than  dimensions. 

3.  Definitions.  A  combination  of  these  multiple  numbers  called  addition 
is  defined  by  the  statement  ,. 

a  +  /3  =  2  (a,  +  h)  e, 

i  =  l 

'HankelI,  Whitehead  1.     Almost  every  writer   has   given  equivalent   definitions.     These    were    of 
course  more  or  less  loosely  stated. 
'  Whiteheab  1. 


10  SYNOPSIS  OF  LINEAR  ASSOCIATIVE  ALGEBRA 

In  quaternions  and  space-analysis  the  definition  is  derived  from  geometrical 
considerations,  and  the  definition  used  here  is  usually  a  theorem.^ 

4.  Theorem.    From  the  definition  we  have 

a  +  (3  =  i3  +  a  a  +  {(3  +  y)  =  {a  +  i3)  +  y 

when  these  equations  hold  for  the  range  of  coordinates.  If  subtraction  is 
defined  for  the  range,  it  will  also  apply  here. 

5.  Theorem.    If  m  belongs  to  the  range  and  if  ma  is  defined  for  the  range 
(called  multiplication  of  elements  of  the  range)  then  we  have 

r 

OT  a  =  2  (m  a,)  Cj 

1=1 

6.  The  units  are  called  units"-,  or  Haupteinheiten'^ ,  and  the  region  they 
define  is  also  called  the  ground^  or  the  hasis^  of  the  algebra.     The  units  are 

written  also"  (1,0,0, ),  (0,  1,  0, ), (0,0, 1),  the  position  of  the  1 

serving  to  designate  them.  The  implication  in  this  method  of  indicating  them 
is  that  they  are  simply  ordinary  units  (numbers)  in  a  system  of  «-tuple  numbers, 
the  coordinates  of  each  n-tuple  number  being  independent  variables.  This  view 
may  be  called  the  arithmetic  view  as  opposed  to  that  which  may  be  called  the 
vector  view,  and  which  looks  upon  the  units  as  extraordinary  entities,  a  terra 
due  to  Cayley.  There  are  two  other  views  of  the  units,  namely,  the  operator 
view,  and  the  algehraic  view.  The  first  considers  any  unit  except  ordinary 
unity  to  be  an  operator,  as  (—1)  or  the  quaternions  i,j,  h.  The  second  con- 
siders any  unit  to  be  a  solution  of  a  set  of  equations  which  it  must  satisfy  and 
as  an  extension  of  some  range  (or  domain,  pr  field);  or  from  a  more  abstract 
point  of  view  we  consider  the  range  to  be  reduced  modulo  certain  expressions 
containing  the  so-called  units  as  arbitrary  entities  from  the  range.  Thus,  if 
we  treat  algebraic  expressions  modulo  i'^  +  1,  we  virtually  introduce  V —  1 
into  the  range  as  an  extension  of  it.'^ 

7.  Definition.    We  may  now  build  a  calculus**  based  solely  on  addition  of 
numbers  and  combinations  of  the  coordinates.     This  may  be  done  as  follows  : 

Let  the  symbol  /  have  the  meaning  defined  by  the  following  equations  :  if 

r  r 

a  =  2  Oi  ej  ^  =  2  Xi  €i 

1=1  i=l 

then 

r 

/  .  a  ^  =  2  UiXi 

i  =  1 

It  is  assumed  that  the  coordinates  a,  x,  are  capable  of  combining  by  an  associ- 
ative, commutative,  distributive  process  which  may  be  called  multiplication, 
so  that  UiXi,  is  in  the  coordinate  range  for  every  Oj  and  a-j,  as  well  as  XafCi. 


"Hamilton  1,  2,  Grasshann  1,  2,  cf.  Macpablane  1.  'Grassmann  1. 

«  Weiekstrass  2.  «Taber1.  SMoLiENl.  «Dedekind  1,  Bkki-ott  1. 

iguAW  13.  *See  §21  for  dillereuce  between  a  calculus  aud  an  algebra. 


DICFIiNlTIONS 


11 


Evidently 


1  .  ri^  =  x. 


E=lcJ.e,^, 


(    1 


Also,  if  /  ::}:./ 

/.f'.^'i=l  1.6(61=0 

8.  Theorem.    We  have 

9.  Definition.    We  suy  thut  a  and  ^  are  orthogonal  if  7  .  a^  =  0.     The  units 
g].  .  .  .,  Cr  therefore  form  an  ortJto(jonal  system. 

If  /  .  ^^  =  0,  ^  is  called  a  nullilat. 

10.  Theorem.    Let 


and 


Then,  we  have 


I .  E,E,=  \ 


{i=l....r) 


k«r'=l 


Inhere  C'y  is  the  minor  of  Cj,  in  \c 
Further 


a- 


la,  a,  E, 


If  /'.  refers  to  the  E  coordinates  just  as  /to  those  of  the  e's, 

r 

since  2   Cy  C,/,  =  0  or  1  as  k  :^j  or  /.•  =j,  and  Jcy  |-  =  1. 

i=  1 

Hence  /  is  invariant  under  a  change  to  a  new  orthogonal  basis. 

11.    Definition.    Let    the  expression  A  .  a^  ....  a,„_i  Afi^.  .  •  ./3,„  represent 
the  determinant 

/3l  /?2  ^3 /?. 


/a,„ _  1  /?i     Ta,„  _  1  /?o      /a„,  _  i  /^j  ....    /x,„ _  j  /3„ 

In  particular 

^i  .  Uj  J-p'i  /Sa  =  /?j  /a,  /?2  —  iSg  /tti  /?! 

i  .  aj  ^a,  ^,(^1  ((^^3  =  l^«i  A,  ^asi^s!  =  —  /•  ao^Uj  ^/Jj/^a 
=  /.  {SiA^.y  -4aia3 

These  expressions  vanish  if  ai  .  .  .  .  a„,_i  are  connected  by  any  linear  rela- 
tion, or  /!?i  ....  /3,„  by  any  linear  relation,  or  if  any  a  is  orthogonal  to  all  of 
the  /?'s.     If  any  (3,  say  ^^,  is  orthogonal  to  all  the  a's, 

/  .  tti  Aa.,  ■  ■  ■  ■  a,„  A^i ....  /3„,  =  0 
and 

Aai a,„  _  1  Al3i 3„  =  iSi  /ui  Aa., a„ _i  AjSo  ■  ■  ■    (3„, 


12 


SYNOPSIS  OF  LINEAR  ASSOCIATIVE  ALGEBRA 


12.  Theorem.  A  .  aA^y  +  A  .  ^Aya  +  A  .  yAa^  =  0 

/ .  aA^Ayh  +  I .  (SAyAab  +  I .  yAaAi3h  =  0 

j3I  aa  =  a/a/3  —  AaAa^ 
A  .  aia2A(3i(3.,(3a  =  ^iI.a,AaJ(32l33  —  (S,J.a^Aa2A(3i(i-i  +  t^3l.aiAa,A(3i^-. 
=  Aa,  A13, 13,  la.,  ^3—^  ai^/3i/?3  ^^a^A  +  A  a^A/SoJ^  la^^y 
I.  ayAao  as  A(3i  /J,  /?3  =  —  ^  '  0.2  -^Ci  «3  -^/^i  /^2  /^3  =  •  •  •  • 

=  I. l3iAl32 ^3  Aaiasa3=  —  I.  /?„  l/3i/?3  iajaoag  =    ... 
y  I .  aA(3Aa^  =  al .  aA^Ay^  +  ^I.  aA^Aay  +  Aa^Aa^y 

13.  Theorem.    In  general 

A.ai-  .  ■  .a„i_i-4/?i.  .../?„  =  2  .  (SiIa^Aa,-  ■  ■  .a,^^iA(3o.  •  ■  •/?„, 

=  2  .  Aa,  A^if^z  /ttg  Aas .  .  .  .  a„_i  A^a ..../?„ 

=  2  .  AaiaoA^ilB^Sal.asAai.  ■  ■  .a„_i  A/Si-  .  .  .(3^ 

(31.  aiAao-  ■  ■  ■a„_i-4ai.  .  •  .a„_i  =  2  .ai/ai^ag-  •  •  -ar^^iAfia,-  ■  ■  -an..! 

Signs  of  terms  follow  rule  for  Laplace's  expansion  of  a  determinant.    Develop- 
ments for  J-ttj  A(3iy  and  higher  forms  are  easily  found. 


14.    Theorem.    If  the  notation  be  used 
^1^12  ••*j  =  ^  .:^,A.AX,....A.  A?.,A(io^,.fi,_ 


I i"2  •  1«1 


then 


i«0  i«l  /"3  /"3 ^.-1  f's 

/X3//0  0  0  /Jl3i>'3 I^3f^s-1  ^^3(^B 


I\  ."0 


-^^s^« 


It  follows  that 

A   .   X1X2  A  (1q  jUi  |M2 


=  ^fj»)-/.X,„.^{j;} 


DEFINITIONS  1  .'J 

Omitting  X  and  fi 

f  ]  23 )  (  2'>1  f  ■■'3  ) 

The  forms  J. .  .  .  .  yl .  .  .  .  may  ail  be  developed  in  this  manner. 

The  form  AV^l^         V  I ,   where   i, i„,  Ji /„  are  two  sets  of  n 

subscripts  each  chosen  from  among  the  r  numbers  1  .  .  .  .  r,  may  be  looked 
upon  as  determining  a  substitution  of  n  cycles  on  the  r  numbers,  the 
multipliers  J'/.j^^^fii^^^^,-  ■  ■  -I'/.j^Hi^  furnishing  the  other  r — n  numbers,  that  is, 
the  whole  term  determines  the  substitution 

J  *)i  +  l  ■  ■  ■  •   trj1'\  ■   ■  •  •    K   I 
(^ri+1  •       •  •  JryJl  ■  ■  •  ■  Jn) 

which  must  contain  just  n  cycles.  It  is  also  to  be  noticed  that  ii^jt, 
t^  1  .  ...  71.  The  terms  in  the  expansion  of  J. .  Xj  .  .  .  ./.^  -^/«o,«i  ■  ■  ■  ■  f^r  are 
then  the  r!  terms  corresponding  to  the  r!  substitutions  of  the  symmetric  group 
of  order  r\.  The  sign  of  each  term  is  positive  or  negative  according  as  the 
number  of  factors  /  in  front  of  the  A  ]  \  is  even  or  odd.  Certain  theorems 
are  obvious  consequences  but  need  not  be  detailed, 

15.  Definition.    Let  §(a/3)  be  any  expression  linear  and  homogeneous  in 
the  coordinates  of  a  and  (3. 

Also  let 

be  formed.     This  is  called  the  Q-th  bilinear  ^.^ 

16.  Theorem.    If  e,'  is  any  other  orthogonal  system, 

Q.^^  =  :iQ.cjel.Iejet  He, 

^  y   •  ^i  6« 

Hence  Q  .  ^^  is  independent  of  the  orthogonal  system. 
It  follows  at  once  that 

I .  ^A?.i  A^fii  =  (r  —  1 )  /Xi  [li  A  .  p.i  A^fii  fi,  =  —  (;•—  2)  ^;Li  Au,  ^ 

/ .  ^A2.i .  .  .  .  X,  A^u^ .  .  .  .  Uj,  =  (r  —  s)  /.  Jlj  Ax, .  ■  .  ■  '/.^  Aui ...._«, 

J. .  ^Xj     ?i,  A^f^^  .  .    .  j«,+i  =  —  (r  —  s  —  1 )  .4X1 /.,  Au^ ju,+i 

Q  .  ^^  may  also  be  written  Q  .  y^  by  extending  the  definition  of  v>  ^he 

coordinates  of  ^  being  x^  .  .  ■  ■  x,,  that  is,  V  =  2  C;  -^— . 

'  M'AlLAT  1. 


14  SYNOPSIS  OF  LINEAR  ASSOCIATIVE  ALGEBRA 

17.    Theorem.    By  putting  subscripts  on  the  zeta-pairs  we  may  use  several. 


Thus 

A.^,^,Ap^,^C2=     (r-2)(r-l)p 
/  .  ^1  A^,  ;ii  A(,  ^3  ^,1=     ir~2)  [r  -  1)  I  A,  ^i, 

A^,  ^2  ;ii  A^i  ^2  ^1  ^u  =     (r  -  3)  {>■  —  2)  4X,  Afi,  fi^ 
In  general 

I^,  A^,...^,A^,...^,  =  r{r-l)...ir-s  +  1) 

7^1  ^^o. . .  g  ;ij . . .  X,  A^,  ...^,fi,...^i,= 

(r_,s)  (r-s-  i)...(r-s  — <+  1)  /.  ^1  A?.,...?.,  A«i  •  •  •  i"» 
If     s  +  <  >  r  this  vanishes ;   i£  s  -\-  t  =  r,  yve  have 
7  .  /^j  ^/lo .  .  .  /Ij  Af.li  ■  •  ■  f's  ^ 

^^1 . . .  ^j  Xj . . .  ?-s_i  .4{;i . . .  ^'^  j(fi . . .  fi,  = 

(-1)'  (r  -  6-) .  .  .  (r  _  5  —  <  +  1)  .4^)  .  .  .  ;i,„,  ^,«i ...//, 
All . . .  ?u,_i  Afii . .  .  fi,— 

18.    Theorem.    If  7cc,  p  =  0         i  z=  i  .  .  .  .  m  —  1,         then 

p=  ^  .  «!•  .  .  -a^.i  J/;?j /3„ 

where  /3j,  (/==!....  m)    is    arbitrary.      For,  if    we    take    the   case    where 
m  —  1  =  3,  we  have  for  /3],  /J^,  /?a  all  arbitrary,  the  identity 

J«,  ao  ttj  ^4/3i  (i..  ^ip  =  /?!  /ai  -la,  aj  -4/33  /^s  P  —  /^3  -^"i  -^'^'s  «3  -^/^i  /^3  P 

+  /^a  7/1  ^a.  ag  J/^j  /^o  p  —  p/aj  Aa^  "3  ^/3i  ^^  /^g 
Hence 

p  7«i  yiaa  a-j  A(3i  ^^  ^3  =  (3^  7«,  Aa-,  a^  Afi.^  /?3  P  —  /^g  -^"1  ^aa  "s  ^i^i  /?3  P 

+  /^g  7ai  -(4rx3  Kg  J./3]  Z?,  p  —  -4a]  ao  a^  A(3i  ^.,  /^g  p 

Since  7a)  p  ==  0         /'/.2P  ^^  0         -^"sP  ^  0  i  therefore  identically 

7a] ,/?]  Txi  ^a^ag  A^.^fS^p  —  Ia^[3.,IaiAaoa:tA^i  i5gp  +  lixS^Ia.iA'XMj  -^/^i/J^p  =  0 

with  two  similar  equations  fur  a.,,  Wg.     Therefore,  since  /3],  /^o,  .<i?3  are  arbitrary 

7ai  Aa^.  ttg  ^/3i  /^.,  p  =  0  7a]  Ja^  ag  ^1/3]  /i.,  p  =  0  /a,  ylaj  ag  Aj^.^  /3g  p  —  0 

or  else,  for  any  /3j,  /?3,  /?g, 

/aj  Aa.^  ag  -4/3]  /^o  /?3  =  ^ 


DEFINITIONS  15 

This  is  impossil)le,  lience 

p  /ui  Aa.^  (la  A(3i  (3..  /?3  =  —  via,  a.. «;,  AfSi  (I,  /?«  p 

or  p  =  -4«i  a.,  «3  -4/:?j  /?3  /iy  [ii 

where  /?,,  /?a,  /?;,,  ^^  are  arbitrary.  A  similar  proof  holds  for  the  general  case. 
This  ciilciiliis  would  enable  us  to  produce  a  tlieory  of  all  bilinear  functions 
(2(ap),  and  thus  the  so-called  algebras.* 

19.  Definition.  A  subregion^  consists  of  all  hypercomplex  numbers  which 
can  be  expressed  in  the  form  a  =  «i  p,  -f-  <';•  P;;  +  •  •  •  •  +  "iPi  wherein  p,,  p^, 
....,  Pi  are  given,  linearly  independent,  numbers  of  the  range  of  the  algebra. 

20.  Theorem.  An  unlimited  number  of  groups  of  m  inilependent  numbers 
can  be  found  in  a  region  of  m  dimensions.'^  Any  group  is  said  to  define 
the  region. 

21.  Definition.  Tlie  calculus  of  these  entities  is  called  an  uhjehrd,  if  it 
contains,  besides  addition,  another  kind  of  combination  of  its  elements,  called 
multiplication.  The  algebra  is  said  to  beef  finite  dimensions,  when  it  depends 
on  r  units,  r  being  a  finite  number.  Of  late  the  term  finite  has  been  applied 
to  algebras  the  range  of  whose  coordinates  consists  of  a  finite  number  of 
elements. 

Multiplication  is  usually  indicated  by  writing  the  niuiibers  side  b}'  side, 
thus,  a^  or  a. /3.  Upon  the  definition  of  multiplication  depends  the  whole 
character  of  the  algebra.*  The  definition  usually  given  is  contained  in  the 
statements : 


if 

a  =     2     «;  Cj 

k=l 

then   a.  ^  ^  y 

if 

l....r 

c,=    1    .  or,  .  hj  . 

Yijk 

{Jc=l,2....r) 

The  constants  j/;^^.  are  called  constants  of  muUipUcation.  If  tnultiplication 
is  defined  in  this  manner  the  algebra  is  called  linear.  The  products  o, .  hj  are 
defined  for,  and  belong  to,  the  range  of  coordinates.  The  constants  of  mul- 
tiplication also  belong  to  the  range,  and  their  products  into  a,  t,  are  defined 
for,  and  belong  to  the  range.  Algebras  whose  constants  are  such  that 
|yJ^^.  =  y^j/.  are  called  reciprocal.     If  y,'j^=  yfj^,  they  ViVQ  parastropldc. 

22.    Theorem.    If  multiplication  is  defined  as  in  §  21,  then 

a  .  (/?  +  y)  =  a  .  /3  +  a  .  7  {a  -\-  ^)  .  y  =  (x.  y  +  ^  .  y 

{a^-  ^).{y  +  h)  =  a.y  ^a.h-V  ^.y  +  [i.h 
This  is  usually  called  the  distributive  law  of  multiplication  and  addition.     An 
algebra  may  be  linear  without  being  distributive.^ 

'Whitbheab  1,  p.  123.  2Cf.  Whitehead  1,  p.  123.  ^cf.  Gibbs  2,  Macfarlase  4,  Shaw  1. 

*Shaw  9.  6  Dickson  7. 


1  6  SYNOPSIS  OF  LINEAR  ASSOCIATIVE  ALGEBRA 

23.  Definitions.  In  the  product  a/?,  a  is  called  the  facient,^  or  the  le/t 
factor,  or  the  pref actor  f  /?  is  called  the  faciend,^  or  right  factor,  or  post/actor.'^ 
The  latter  names  will  be  used  in  this  memoir. 

If  there  is  a  number  a^  in  the  algebra,  such  that  for  every  number  of  the 
algebra,  a,  a  ag  =  a  =  a^a,  then  ao  is  called  the  modulus^  of  the  algebra. 

If  we  have  a  ao  =  a         aja  ^^  a  we   may  call  a^  a  post-modulus. 

If  we  have         Oq  a  =  a         a  .ao  :|:  a  we  may  call  Oo  a  pre-modulus. 

In  defining  an  algebra,  the  existence  of  a  modulus  may  or  may  not  be 
assumed.  When  for  all  numbers  a,/?,  we  have  a ^  =  13 a,  the  algebra  is 
called  commutative. 

When  for  any  three  numbers  a,  /3,  y  we  have  a  .{^  .y)  =  {a  .  (i) .  y, 
the  algebra  is  called  associative.* 

24.  Theorem.  If  an  algebra  is  linear,  the  product  of  any  two  numbers 
is  known  when  the  products  of  all  the  units  are  known.  These  products 
constitute  the  midliplicafion  tabic  of  the  aigebra. 

25.  Theorem.  In  an  associative  algebra  the  constants  of  multiplication 
satisfy  the  law 

r  r 

2   ra«  y,jt  =    2   y;,js  y^t  {i,  1^,j,  t  =  1 r) 

s=l  s=l 

26.  Definitions.    If  a  .  a  ^  a"  =  a,  then  a  is  called  idempotent. 

If  a'"  =  0,  m  a  positive  integer,  then  a  is  called  niljwtent,  of  order^  wj  —  1. 
If  a, 6'  =  0,  then  a  \s  p)re-nilfactorial  to  /3,  which  is  post-nilfacturial  to  a. 
Ifa/i  =  0  =  /^a,  then  a  is  nilfactorial  to  |3,  and  (i  to  a. 

27.  Definition.  The  expression  I.  a  (3  is  sometimes  called  the  inner  or 
direct pjroduct^  of  a,  /3  and  written  a*  /3.     Further,  the  expression 

Q{a^)  =  Y  a,bj  .e,Iej{) 

is  called  the  dyadic  of  a  (5,  and  written  a  /?.  It  is  thus  an  operator  and  not  a 
product  at  all.  The  use  of  the  term  product  in  similar  senses  is  quite  common 
in  the  vector-analysis,  but  it  would  seem  that  it  ought  to  be  restricted  to 
products  which  are  of  the  same  nature  as  the  factors.  Gibbs,  however,  insisted 
that  any  combination  which  was  distributive  over  the  coordinates  of  the  factors 
was  a  product.' 

There  is  no  real  difference  between  the  theories  of  di/adics,  matrices,  linear 
vector  operators  J  bilinear  forms,  and  linear  homogeneous  substitutions,  so  far  as  the 
abstract   theory  is   concerned   and  without  regard  to  the  operand."      If  we 

'  Hamilton  1,  B.  Peirce  3.  'Taiieii  5. 

^ScuEFFKKS  1,  Study  1,  who  calls  it  one  (Eins),  identifying  it  with  scalar  unity.  Some  call  it  Ilaupt. 
einbeit.     Cf.  Shaw  1. 

<B.  Peihce3.  'B.  FeikceS.  «  Gibbs  3.  '  Giuns  3. 

*Fkobenius  1,  and  any  bibliography  of  matrices,  bilinear  forms,  or  linear  homogeneous  Bubstitutlons. 
Cf.  Laurent  ],  2,  3,  4.     See  Chap.  XXX  this  memoir. 


DEFINITIONS  17 

denote  the  operator  Q  (a/3)  by  4>,  then  the  bilinear  form  Scy  x,  yj  may  be  written 
J.otpa  or  I.a^p,  where  <^  (or  ^')  is  called  the  conjugate,  the  trnnsvertie,  or  the 
transpose  of  4).  Besides  the  ordinary  combination  of  these  operators  by 
"multiplication"  Stki'IIANos*  defines  two  other  modes  of  composition  which 
may  be  indicated  as  follows  in  the  notation  developed  above : 

(1)  Bidlternafc  composition  in  which 

<^i .  <^2  is  equivalent  to  „  ,  C\.,  Tp'Ap"  A<pi  a'  ^o  a" 

<?>i  •  <?>2  ••■•<?>«  is  equivalent  to  -^|  6j     ,  I^'Ap" ....  p'"*  A(pi  a' ....  <^,  a" 

C,.  ,  indicates  that  the  sum  is  to  be  taken  over  all  terms  produced  by  permut- 
ing in  every  way  the  subscripts  on  the  ^'s. 

(2)  Conjunction,  which  corresj)onds  to  the  multiplication  of  algebras, 
and  is  equivalent  to  taking  <^i  and  4)3  on  different  independent  grounds 
ei  ....  e^,  e[  ...  .  e^,,  whose  products  Cje,'  define  a  new  ground 

fiy  =  Cjc;  {i=l  ..  ..  r,j  =  I  ....  /) 

Thus  <^i  X  (?)a  = '  '2  '  '  2 'eg'  cf}  ea,  Teji 

i.j  k.  I 

2.   definitions  by  independent  postulates. 

28.  Definition.  Three  definitions  by  postulates  proved  to  be  independent 
have  been  given  by  Dickson."     The  latest  definition  is  as  follows: 

A  set  of  7j  ordered  marks  a^  ....  a^  o?  F  (a  field)  will  be  called  an  n-(nple 
element  a.  The  symbol  «  =:  (aj  ....  a,.)  employed  is  purely  positional,  with- 
out functional  connotation.  Its  definition  implies  that  a  =■  h  if  and  only  if 
ttj  ^  6,  ....  a,.  =  b,. . 

A  system  of  «-tuple  elements  a  in  connection  with  n^  fixed  marks  yij^ 
of  i*' will  be  called  a  closed  system  if  the  following  five  postulates  hold. 

Postulate  I:  If  a  and  h  are  any  two  elements  of  the  system,  then 
«  =  (oj  +  6,  ....  a,  +  h^)  is  an  element  of  the  system. 

Definition  :     Addition  of  elements  is  defined  by  a  ®  6  =  s. 

Postulate  II:     The  element  0  =  (0  ....  0)  occurs  in  the  system. 

Postulate  III:  If  0  occurs,  then  to  any  element  a  of  the  system  corre- 
sponds an  element  a'  of  the  system,  such  that  aea'  =  0. 

Theorem :     The  system  is  a  commutative  group  under  9. 

Postulate  IV:  If  a  and  h  are  any  two  elements  of  the  system,  then 
P  =  {jh  •  •  •  •  2\)  is  an  element  of  the  system,  where 

l..r 

Pi  =    2    ajb^  yj„i  (j  =  1 r) 

Definition:     Multiplication  of  elements  is  defined  by  a  ®  6  =^. 

'  Stephanos  6.  'Dickson  5,  8. 


18  SYNOPSIS  OF  LINEAR  ASSOCIATIVE  ALGEBRA 

Postulate  V:     The  fixed  marks  y  satisfy  the  relations 

r  r 

2  r^tj  yji-i  =   2  y,  ij  y,ji  (s,  t,k,i=  1 r) 

J=l  3=1 

Theorem:     Multiplication  is  associative  and  distributive. 

Postulate  VI:  IfTj  ....  T,.  are  marks  ofii^such  thatTjrtj  +  •  •  ■  +T,.flr:=  0 
for  every  element  (oj  ....  o,)  of  the  system,  then  t^  =  0  .  .  .  .  t^  =  0.  [This 
postulate  makes  the  system  r  dimensional'^. 

Theorem  :  The  system  contains  r  elements  s^  =  {(l^■^  ....  a.-r),  i  =  1  .  .  .  .  r 
such  that  I  fly  I  -^  0. 

Theorem:  Every  ?--dimensional  system  is  a  complex  number  system. 

Generalization:  If  the  marks  a^  .  .  .  .  a,^  belong  to  a  field  F^;  and  if 
flr,  +  1  •  •  •  •  (^r,+r.  beloug  to  a  field  F^;  .  .  .  . ,  if  a  corresponding  change  is  made 
in  postulate  VI ;  if  further  yj,,i  =  0,  when  j,  k,  i,  belong  to  different  sets  of 
subscripts,  then  we  have  a  clcsed  system  not  belonging  to  a  field  F} 

3.     DEFINITIONS   IN  TERMS  OF  LOGICAL  CONSTANTS. 

29.  This  definition  is  recent,  and  due  to  Bertrand  Russell.  By  logical 
constants  is  meant  such  terms  as  class,  relation,  transitive  relation,  asymmetric 
relation,  ichole  and  ^)a?-<,  etc.  Complex  numbers  are  defined  in  connection 
witli  dimensions,  or  the  study  of  geometry .  The  definition  in  its  successive 
parts  runs  as  follows  : " 

30.  Definition.  By  real  number  is  meant  any  integer,  rational  fraction, 
or  irrational  number,  defined  by  a  sequence.  These  have  been  discussed 
previousl}',  in  the  work  referred  to. 

A  hypercomplex  number  is  an  aggregate  of  r  one-many  relations,  the 
series  of  real  numbers  being  correlated  with  the  first  r  integers.  Thus,  to  the 
r  integers  we  correlate  flj,  o,  ....  a,.,  all  in  the  range  of  real  numbers.  This 
correlation  is  expressed  by  the  form 

The  order  of  writing  the  terms  may  or  may  not  be  essential  to  the  definition. 
The  e  indicates  the  correlation,  thus  Cj  is  not  a  unit,  but  a  mere  symbol,  the 
unit  being  le^.  The  remaining  definitions,  addition,  multiplication,  etc.  may 
be  easily  introduced  on  this  basis. 

Theorem :  Hypercomplex  numbers  may  be  arranged  in  an  r-dimensional 
series. 

31.  A  like  logical  definition  may  be  given  when  the  elements  belong  to 
any  other  range  than  that  of  "real"  numbers. 

4.     ALGEBRAIC  DEFINITION. 

32.  The  preceding  definitions  are  of  entities  essentially  multiplex  in 
character.     The  units  either  directly  or  implicitly  are  in  evidence  from  the 

'Cf.  Cakstens  1.  «B.  Russell  1,  pp.  378-379. 


DEFINITIONS  19 

beginning.  It  seems  desirable  to  avoid  this  multiplicity  idea,  or  implication, 
until  tbe  development  itself  forces  it  upon  ns.  Historically  this  is  what  hap- 
pened in  Quaternions.  Originally  (|uaternions  were  operators  and  their 
expressihility  in  terms  of  any  independont  four  of  tlieir  nnniber  was  a  matter 
of  deduction,  while  Hamim'oi^  always  resisted  the  coordinate  view.  The  fol- 
lowing may  be  called  llic  algebraic  definition,  since  it  f(dl()\VK  the  lines  of 
certain  algebraic  developments. 

33.  Definition.  Let  there  be  an  a.ssemblage  of  entities  ^(,  either  finite  or 
transfinite,  enumerable  or  non-enumerable.  They  are  however  well-delined, 
that  is,  distinguishable  from  one  iinothcr.  Further,  let  these  entities  be  subject 
to  processes  of  deduction  or  inference,  such  that  from  two  entities,  A^,  Aj,  we 
deduce  by  one  of  these  processes,  passing  from  -4j  to  Aj,  the  entity  A,/,  which 
we  will  indicate  by  the  expression 

Ai  OAj  =  A/,  {Ai,  Aj  any  elements  of  the  assemblage) 

A  different  process  0'  would  generally  lead  to  a  different  entity  A',,;  thus 

A,  a  A;  —  Al 

(These  processes  may  be,  for  example,  addition  °d,  and  multiplication  0).  It  is 
assumed  that  these  processes  and  their  combinations  are  fully  defined  by 
whatever  postulates  are  necessary.  Then  the  entities  A,  and  the  processes 
O,  0' .  .  ■  ■  are  said  to  form  a  calculus,  and  the  assemblage  of  entities  will  be 
called  a  range. 

34.  Definition.  Let  there  be  given  a  range  and  its  calculus,  and  let  us 
suppose  the  totality  of  expressions  of  the  calculus  are  at  hand.  In  certain  of 
the.se,  i^/l,  M.,.  .  .  .M^,  let  us  suppose  the  constituent  entities  Ai,  Aj  .  .  .  .  are 
held  as  fixed,  and  that  we  reduce  the  totality  of  expressions  modulo  these 
expressions  M\  that  is,  wherever  these  expressions  occur  in  an}-  other  expression, 
they  are  cancelled  out.  Then  the  calculus  so  taken  modulo  M  is  called  an 
algebra. 

For  example,  let  the  range  A  be  all  rational  numbers.  Let  the  expres- 
sions iV  be  r  •     .    -,  ^ 

Then  an  expression  like  4  —  8  may  be  written  4i -|-  4  +  4  —  8  =  4i;  an 
expression  like  x~  -f  9  becomes  ar  +  9  —  (9  -f  9^")  =  x-  —  9^- ;  which  may  be 
factored  into  {x  +  sy)  (a;  —  Zj)  or  (x  +  3^)  (x  +  Sy ). 

In  this  manner  we  have  a  calculus  in  which  will  always  appear  the 
elements  i,  j  {or  J  andy-  as  we  might  find  by  reductions).  Modulo  i  +  1  and 
y"-f  1,  certain  expressions  become  reducible,  that  is  factorable,  which  other- 
wise cannot  be  factored.  We  call  the  expressions  xi,  xj,  xj^,  in  this  case, 
where  x  is  any  rational  number,  negative  numbers,  imaginary  numbers,  and 
negative  imaginary  numbers.  We  consider  i  andy  as  qualitative  units,  although 
perhajDS  modular  units  would  be  a  better  terra. 


20  SYNOPSIS  OF  LINEAR  ASSOCIATIVE  ALGEBRA 

35.  It  is  not  assumed  necessarily  that  there  is  but  one  entity  A^  for  any 
given  expression,  for  we  may  have  two  expressions  alike  except  as  to  the 
elements  that  enter  them.     Thus  we  might  have 


"^e' 


31 


l;'::} 


36.  Definition.  In  any  case  we  shall  call  the  expressions  31  the  defining 
expressions  of  the  algebra,  and  the  elements  A^  (such  as  i,j)  entering  them  the 
fundamental  qualitative  units. 

37.  Postulates  : 

I.    It  is  assumed  that  the  processes  of  the  calculus  are  associative. 
II.    It  is  assumed  that  the    processes    which   shall  furnish   the  defining 
expressions  shall  be  those  called  addition  e,  and  midtiplicatiun  ®. 

III.   It  is  assumed  that  the  process  ®,  multiplication,  is  distributive  as  to  the 
process  ®,  addition.     Tiiat  is 

Ai  ®  {Aj  ®  A^)  =  {A,  ®  4)  ©  {A^  ®  A) 
{Aj  ©  A^)  ®  A,    =  {Aj  ©  A-)  ©  (A^.  ®  Ai) 

38.  The  comrautativity  of  multiplication  is  not  assumed.  Further,  the 
general  question  of  processes  and  their  relations  is  discussed,  so  far  as  it  bears 
on  these  topics,  in  XIII,  hence  will  not  be  detailed  here. 

It  is  evident  according  to  this  definition  that  an  algebra  may  spring  from 
an  algebra.  Hence  the  term  is  a  relative  one,  and  indeed  we  may  call  a  cal- 
culus an  algebra  if  we  consider  that  the  calculus  is  really  taken  modulo 

A,OA^  —  A,  A,0'Aj-M,  etc. 

That  is,  the  equalities  or  substitutions  allowed  in  the  calculus  make  it  an 
algebra.  The  only  calculus  in  fact  there  is,  is  the  calculus  of  all  entities 
J.,,  Aj,  Ak,  etc.,  which  permits  no  combinations,  that  is,  no  proces.«es,  at  all. 
From  J.;,  Aj, ....  we  infer  or  derive  nothing  at  all,  not  even  zero.  The  calculus 
of  symbolic  logic  is  thus  properly  an  algebra. 

Any  definition  of  an  algebra  must  reduce  to  this  definition  ultimately, 
for  the  multiplication-table  itself  is  a  set  of  r"  defining  expressions.  That  is, 
we  work  modulo^ 

r 

Cj  gj  —  2  Yijk  ek  (*,  y  =  1   •  •  •  •  r) 

k=l 

39.  Definition.  If  the  range  of  an  algebra  can  be  separated  into  r  sub- 
ranges, each  of  which  is  a  sub-group  under  the  process  of  addition  e;  so  that 
an  entity  which  is  the  sum  of  elements  from  each  of  the  sub-ranges  is  not 
reducible  to  any  entity  which  is  a  sum  of  elements  from  some  only  of  the 
sub-ranges;  then  the  algebra  is  said  to  be  (additively)  r-dimensional. 

iCf.  Kboneckeu  1,  where  this  view  is  very  clearly  the  basis  for  commutative  systems. 


DEFINITIONS  21 

40.  It  iH  to  bo  noted  tlwit  an  algebra  may  be  /■-dimensional  and  yet  have 
in  it  /•  +  A'  distinct  qualitative  units.  ThuH,  ordinary  positive  and  negative 
numbers  form  an  algebra  of  two  units  but  of  only  one  dimension.  Ordinary 
complex  numbers  contain  four  qualitative  units,  but  form  an  algebra  of  two 
dimensions. 

The  defining  exprcs.sions  determine  the  question  of  dimensionality.  For 
example,  let  the  defining  expressions  be 

f  e'i—  I         e'i  —  1         e^Ci  —  e\ e.^ 
t  ei  +  e\  +  1 
whence  we  may  add 

^1  *2  +  ^  ^2  +  <^2  ^3  *1  ^2  ^1  1  ^1  ^2  ^1  ^2 1  Ci  6^  ^  6^  Ci  —    1 

We  have  here  two  more  defining  expressions  than  are  needed  to  define  an 
algebra  of  six  units,  hence  the  algebra  becomes  four-dimensional.  The 
problem  of  how  many  defining  expressions  are  necessary  to  define  an  algebra 
of  r  units  has  never  been  generally  solved  even  for  such  simple  algebras  as 
abstract  groups.  If  the  algebra  is  finite  of  order  r,  a  maximum  value  for  the 
number  is  'r.     But  a  single  expression  may  define  an  infinite  algebra. 

Nothing,  so  far  as  known  to  the  writer,  has  been  done  towards  the  study 
of  these  algebras  of  deficient  dimensionality. 


n.     THE  CHARACTERISTIC  EQUATION  OF  A  NUMBER. 

41.  Theorem.  Any  number  ^  in  a  finite  linear  associative  algebra  which 
contains  a  modulus,  e,,,  and  whose  coordinates  range  over  all  scalars,  satisfies 
identically  an  equation  of  the  form  A'  (^)  =  0,  and  equally  an  equation  of  the 
form  A"  {^)  =  0.  In  each  case.  A'  (^)  or  A"  (^)  is  a  polynomial  in  ^  of  order  r, 
the  order  of  the  algebra.' 

The  function  A'.  ^,  called  the  pre-latent  function"  of  ^,  has  the  form 

2  .  Xi  y.ii  eo  —  (;      2  .  Xf  yi2i  2  •  x,-  >/,■,, 

The  function  A".  ^,  called  the  post-latent  functionr  of  ^,  has  the  form 

2.x,yii3  'S,.XiYov>€Q  —  ^       2.x,.y,.(2 


A" .  ^  = 


S-a-jyiir  S.Xjysir  2 .»,  y^ir  «b— <f 


'The  relation  between  this  equation  and  the  corresponding  equation  for  matrices  is  so  close  that  we 
may  include  in  one  set  references  to  both;  Cayi.ey  3;  LaguerreI;  B.  Peirce  1,S;  Frobenivs  1,2; 
Sylvester  1,3,3;  Bucbbeim  3;  Scuefpers  1,-',  3;  Weyr  1,5,8;  Taher  1,4;  Pascu  1 ;  MoLlEN  1; 
Cartan  2;  Shaw  4. 

^Cf.  Taber  1. 

2 


^2 


SYNOPSIS  OF  LINEAR  ASSOCIATIVE  ALGEBRA 


In  each  case  2  stands  for  2  .     These  functions  may  be  expanded  according  to 


1=1 


powers  of  ^,  taking  the  forms 

A'.^  =^^—m[  ^^-'  +  mL  T"" +  {—Ymle, 

A".^  =  r  -  m['  r-'  +  mi,'  ^'-'^ +  (-)'•  ml' e„ 

In  certain  cases  (viz.,  when  the  algebra  is  equivalent  to  its  reciprocal)  these 
two  become  identical.  (The  absence  of  a  modulus  does  not  add  to  the 
generality  of  the  treatment.)  These  equations  exist  for  all  ranges  of 
coordinates. 

42.  Definition.  The  coefficients  m[  and  wjj'  are  respectively  the  pre-scaJar 
and  the  post-scalar^  of  ^,  multiplied  by  r;  that  is,  if  we  designate  the  scalar 
of  ^  by  S  .^,  we  have 

g,y_.m[  ^„    X  _  »< 

r  r 

If  we  indicate  S.^^  by  Si,  we  have  by  well-known  relations  from  the  theory 

of  algebraic  equations 

rSi  1  0       0      0 

rSo  rSi  2       0      0 

rSs  rSs        rSi     3      0 


m:  = 


Theorem:     The  symbol  S  obej's  the  laws^ 


i—l 


(a,  any  scalar)     *$" .  Co  =  1 


43.  Definition.  The  number  F'.  ^  =  ^  —  S'.  ^  is  the  pre-vector'^  of  ^, 
and  the  number  V".  ^  =  ^  —  /S'".  ^  is  the  post-vector^  of  ^.  By  substituting 
these  for  ^  in  the  identity  for  nij  in  §42  we  arrive  at  various  interesting  and 
useful  formulae. 

44.  Definition.    If  the  two  equations  A'.  ^  =  0,  A".  ^  =  0  are  not  identical, 

the  process  of  finding  the  highest  common  factor  will  lead  to  a  new  expression 

A  .  ^  which  must  vanish.     When  the  two  equations  A'.  ^  =  0,  A".  ^  =  0  are 

identical  we  may  also  have  ^  satisfying  an  equation  of  lower  order;  let  the 

lowest  such  be 

A.^  =  0 

This  single  equation  is  called  the  characteristic  equation  of  ^,  and  A  .  ^  is  the 
characteristic  function  of  ^.*  [The  pre-latent  equation  was  called  the  identical 
equation  by  Cayley,  characteristic  by  Frobenius  and  Molien,  and  this  lower 


■Tabek  2,  3,  4,  G.    Cf.  Frobenius  14,  §4.    Fuouenius  called  m,  the  Spur  of  ^. 

»Tabbii2,  8,  4,  5.  »  Cf .  Taukh  2,  8.  «See  refereuces  to  §41. 


THE  CHARACTERISTIC  EQUATION  OK  A  NUMBER  23 

equation  has  been  called  "  limu/fj/eicJinnf/"  hy  Molikn,  "  GnimJfjkicJturifj"  by 
Wfvk,  idnitlcal  eqwition  and  fuiUhimndal  eqitatioii  by  Tahku,  charadtrihiic 
equation  by  SciiEFFKUS,  and  in  some  cases  it  is  the  reduced  churucleristic 
equation.] 

45.  Theorem.  The  characteristic  function  is  a  factor  of  the  two  latent 
functions.' 

46.  Definition.  The  order  of  the  characteristic  function  being  r';!?-,  it  niny 
be  written 

i^-giCoY' {^—9,.^oYp  1^1  + +  ^.  =  ^' 

The  scalars  7,  ....  </,,  are  the  p  distinct  latent  roots  of  '(.  The  exponents 
Hi  ....  |Up  are  the  j^  suh-mulliplicitiea  of  the  roots  of  ^.  The  factor  ^  —  ^,  Co  'b 
the  latent  factor"  of  the  root  g^. 

Weierstrass  called  (^ — i/i  Co)>  •''•"^  powers,  elementary  factors  {eltmen- 
tartheiler),  particularly  the  powers:  A;^,  { /c^j  _i,i  •  •  •  •  /«i, i-  See  MuTU  1  for 
references  to  this  subject,  or  Weierstrass  1 ;  Kronecker  2,  3,  4;  Frobenius, 
Grelle  86,  88  ;    Berliner  Sitz-her.  1890,  1894,  1896. 

47.  Theorem.  For  a  fixed  integer  i  (1^  i  ^  i)),  there  is  at  least  one  solution, 
a,  {a  -^  0)  for  each  of  the  equations 


A  solution  of  the  k-th  equation  is  a  solution  of  those  that  follow.  If  (T^t 
is  a  solution  of  the  7i;-th  of  these  equations,  then  among  the  solutions  of  the 
k+  1-th  equation,  which  include  the  solutions  of  the  previous  equations,  some 
are  linearly  independent  of  the  entire  set  of  solutions  ct,;,j  of  the  k-th. 
equation.^ 

Theorem :  The  solutions  of  these  equations  for  different  values  of  i  are 
linearly  independent  of  each  other.* 

48.  Definition,    The  number 

y  _  (?  —  ffi  ep)"'  •  •  •  •  (^  —  9i-i  eoYi-i  (?  —  9i+i  epYi+i .  ■  ■  •  (^  -  gr^  e^Yl 

'~l9i~9iY'    •■■   {9i-gt-iY^-'     {9i—9i+iY^^'     ■■■■{9i—9pY^^ 

is  the  i-th  latent  of  ^;  it  corresponds  to  the  root  g^.    There  are  thus  p  latents  of  ^. 

49.  Theorems.  The  product  of  Zi  and  any  number  of  the  algebra  is  either 
zero  or  else  it  is  a  number  in  the  region  of  solutions  of  the  equations  in  §47.^ 
We  may  symbolize  this  by  writing  Zi\a\  =  {^i\  The  region  ]^, [  is  called  the 
i-th  pre-latent  region  of  ^.    There  are  correspondingly  post-latent  regions  of  ^. 

'  Taheu  1  ;   \Vevr8;   Molien  1  ;  Frobenius  14.  *  Taber  1  ;     Whitehead  1. 

^TabkrI;  Whitehead  1 ;  Cartas  3.  *Tabee1;  Whitehead!;   Shaw  4.  »  Suaw  4. 


24  SYNOPSIS  OP  LINEAR  ASSOCIATIVE  ALGEBRA 

The  p  latent  regions  of  ^  together  constitute  the  whole  domain  of  the 
algebra.'     It  is  obvious  that  the  Z's  are  such  that  if  i  ^^i, 

Z,Zj=0  {Z,-e,YiZ^=Q 

50.  Theorem."  The  p  pre-(post-)  latent  regions  are  linearly  independent, 
that  is,  mutually  exclusive,  and  together  define  the  ground  of  the  algebra. 
Each  latent  factor  annuls  its  own  latent  region  but  does  not  annul  any  part 
of  any  other  latent  region.  The  i-th  pre-latent  region  may  not  contain  the 
same  numbers  as  the  i-th  post-latent  region.  The  dimensions  of  the  i-ih.  pre- 
latent  region  are  given  by  the  exponent  of  the  i-ih.  latent  factor  as  it  appears 
in  the  pre-latent  equation.  The  pre-latent  equation  contains  as  factors  only 
the  latent  factors  to  multiplicities  jU-,  such  that 

i^iliii  {i=l  ....  p) 

V 

1=1 
Likewise  the  post-latent  equation  contains  as  factors  only  the  latent  factors  to 
multiplicities  ^[' ,  such  that 

ill   =  («i  (i  =  1 V) 

i=l 

51.  Theorem.'  The  pre-  (post)-  latent  region  ]^jf  contains  jU;  sub-latent 
regions  \I,a\,  l^a'h  •  •  •  •,  ]2,v.,[,  where  each  sub-latent  region  includes  those 
of  lower  order,  say  {Xik]  includes  ^2^'}  if  k' <^k. 

The  region  \'S.ik\  is  such  that  (^ — gieof  ^2«}  =  0,  but  in  \Xik\  is  at  least 
one  number  ct,,,  for  which  (^  —  Jj'ieo)*"^  ^a-  ^  0. 

52.  Definition.  For  brevity  let  ^  —  gi  6^  =  61;  then,  in  ]^,j-,  $^''1  annuls 
certain  independent  numbers  which  no  lower  power  of  d^  annuls.  Let  these 
be  Wij  in  number,  represented  by 

sn     ^31 ^Wiii 

Of  course  any  lo^^  independent  numbers  linearly  expressible  in  terms  of  these 
would  answer  as  well  to  define  this  region,  so  that  only  the  region  is  unique. 
Then  each  of  these  multiplied  by  0,  gives  a  new  set  of  ivn  numbers  independent 
of  each  other  and  of  the  first  set.     Let  these  be 

In  general  we  shall  have  for  the  products  by  powers  of  6j  a  set  of  numbers 
linearly  independent  of  each  other, 

7t  =  0  .  .  .  .  |«,  —  1 


0"   ^yi  =  cyi  r*  =  0 


W 


It 


'Taber  1  ;  Whitehead  1  ;  Suaw  4. 

'TabekI;    WuiTEiiEAUl;    Shaw  4;    WeyrB;    Buciiheim  3,  7,  ".». 

'See  preceding  references. 


THE  CHARACTERISTIC  EQUATION  OF  A  NUMBER  25 

The  region  made  up  of,  or  defined  by,  these  numbers  will  be  called  t\ie  Jirst 
prc-sltear  rcfjioH^  of  the  i-th  latent  region.  It  tnay  he  represented  by  ^A'J^'f. 
Let  there  be  chosen  now  out  of  the  numbers  remaining  in  the  i-th  latent  region, 
w.,j  linearly  independent  numbers  which  are  annulled  by  that  next  lower  power 
of  6i,  say  jti,.,,  which  annuls  these  w.^,  numbers,  but  such  that  6'"'-"'  does  not 
annul  them  and  such  that  Sf's  '^  does  not  annul  any  number  which  0f<a  does 
not  also  annul.  These  numbers  and  their  products  by  powers  of  6i  give  rise 
to  the  second  pre-sJiear  region,  {(ii-^  <[  (li) 

We  proceed  thus,  separating  the   i-iU   latent   region  into  C(   shear    regions, 

|Xi',.[, ,  \X^^,    containing   respectively    (iii  =  fx^)    ^^  >'ht, ,  i^ic,  w^a 

linearly  independent  numbers,  with 

Cl 

j=l 

There  is  a  corresponding  definition  for  the  post-regions. 

53.  Theorem.'-  The  pre-  and  the  post-latent  equations  are  (using  accents 
as  before  to  distinguish  the  two  sets  of  numbers) 

n  di^^'n'-'v  =  0  j=\  ....  c[ 

<  =  i 

YiBT"n''"ii=0  j=\  ....  c;' 

54.  Theorem.  If  all  the  roots  gi  vanish,  ^  is  a  nilpotent,  and  for  some 
power  ^  we  have  ^'"  =  0. 

Further,  for  every  number  there  are  exponents  (Xj,  ^",  such  that 

If  ^  and  a  are  of  the  same  character,^  (aa)  then  for  any  power  /«<,,  ^"'^  a 
and  (T  '('"I-    are  nilpotent. 

The  product  may  not  be  nilpotent  if  ^  is  of  character  (a/3)  and  a  of 
character  (/3a).  If  the  product  is  not  nilpotent  the  algebra  contains  at  least 
one  quadrate.  If  an  algebra  contains  no  quadrates,  ^''^  a  and  (T^"*  are  nilpotent 
for  all  values*  of  a  and  fi^. 

55.  Definitions.  When  the  coefficients  in  the  pre-latent  (post-latent) 
equation  vanish  in  part  so  that 

then  ^  is  said  to  have  racuifi/^  of  order  j!^^'.     There  are  «o  zero-roots,  and  one 
or  more  solutions  of  the  equations 

^0  =  0  ^-CT  =  0  ^"'o  a  =  0  (io  =  fil) 

'SiiAw4  -Su\w4.  2See§5(i.  'Cartan  2;  Taber  4.  > Sylvester  1 ;  Taber  1. 


26  SYNOPSIS  OF  LINEAR  ASSOCIATIVE  ALGEBRA 

The  solutions  of  ^ff  =  0  define  the  mdl-region  of  ^.  The  number  of  inde- 
pendent numbers  in  this  region  (its  dimensions)  is  \\\q  first  mtJlity  of  ^,  say  A;j. 
The  h^  independent  solutions  of  ^^a  =  0,  ^g  if  0,  define  the  first  sub-null- 
region  of  ^,  of  second  nullity  h^;  proceeding  thus  we  have^ 

The  vacuity  of  course  is  given  by  the  equation 

^0  =  ^1  + +  ^^^0 

The  characteristic  equation,  it  must  be  remembered,  contains  ^'^''  as  a  factor; 
the  pre-latent  equation  ^'^''',  the  post-latent  i^'^"".  The  partitions  of  /.ig  which 
satisfy  the  inequalities  above  give  all  the  possible  ways  in  which  the  sub-null- 
regions  can  occur. 

56.  Theorem.^  Each  latent  factor,  ^,-,  is  a  number  whose  pre-latent  (post- 
latent)  equation  will  contain  ^f',  and  whose  characteristic  equation  will  contain 
^f .     The  nullities  of  ^j  are  given  by  the  equations 

Ki-n  =  «'i^  +  <«'3i  <  =  Oor  1 


hi        =  ^li  +  *^2i  + +  ^1  w^c,-  u  +  's  «fc<i  h,    «3  =  0  or  1 

hi  =    W^i   -\-   W^i    + +    Wc,i 

The  vacuity  fil  =  /"a  w'u  +  /^,;3  ^'2;  + +  f-i-iu'^ui 

57.  Theorem.    The  number  ^  may  be  written^ 

wherein  the  numbers  xj     4),     (t  =  1 /))  satisfy  the  following  laws  : 

xf     =  Xi        Xi  xj  =0  if  i  :^  j 

Xi%=zQ  =^jXi  ii*+y 

^i  ^i  =  ^i  =  <?».  ^i 

The  numbers  Xi  and  ^^  are  all  linearly  independent  and  belong  to  the 
algebra,  at  least  if  we  have  coordinates  ranging  over  the  general  scalar  field. 

58.  Theorem.*    Let  ha  ^^  + +  /',>..    1  <?>r'    '  =  ^o   then  if  F  (x) 

is  any  analytic  function  of  cc,  F'  (x) its  derivatives, 

jr,'^=i^S^F{g^).x,-\-F'{gd.%+  ^^^^^  + |~ /)f  *^''"'} 


'8rLVESTER2;  Tabku  1 ;  BuouueimS;   Whitehead  1. 

!§52:  3 Study  G;  Shaw  7.  <Siiaw7.     ('f.  Taiiek  1  ;    Svlvustek  8. 


THE  CHARACTERISTIC  EQUATION  OF  A  NUMBER  27 

59.  Theorem.    Tlie  (liirerent  number.s  of  the  algebra  will  yield  a  set  of 
idempoteut  expressions  e,  ....  c^,  such  that  if  i  ij:/,     i^  j  =  i   ....  a 

^'/  =  ei  efej=0  —  CjCi  gg  =  e,  + -\.  e^ 

and  hence  the  numbers  of  the  algebra  may  be  divided  into  classes  \Z^p\,  such 
thiit  if  ^,.3  is  in  the  cla.ss  \Z^^\,  then 

The  subscripts  a,  /?  are  the  diameters^  (pre-  and  post-  resp.)  of  ^„^.     In  this 
and  similar  expressions  ^j.y  =  0  when  x  -^  1/,  S'^,,  =  1  when  x  =y. 

60.  Theorem.  The  product  of  lif^^  and  ^^j  is  given  (when  it  does  not  vanish 
on  account  of  properties  not  dependent  on  the  characters)  by  the  equation^ 

The  numbers  ^„„  form  a  sub-algebra,  (a  =  1,  .  .  .  .,  a). 

61.  Theorem.  Let  the  characteristic  equation  of  ^  have  q  —  1  distinct  roots 
which  are  not  zero,  and  let  v  —  1  be  the  lowest  power  of  ^  in  this  equation. 
Tlien  if 

we  have'' 

i=l 

<J-1  q-1 

62.  Theorem.*  If  e^  rf:  ^  x^,  then  e„=  2  Xi  +  x,,,    where  x,  belongs  to 

i=l  i=l 

the  root  zero  and 


Theorem :    It  also  follows,  that,  if 


^^=l<-(|E^yr' 


then  i'',  ^  =:  ;Cj 

63.  The  use  of  the  two  sets  of  idempotents  of  ^,  the  pre-  and  the  post-, 
enables  us  to  find  partial  moduli,  which  are  not  necessarily  invariant,  and  the 
modulus,  which  is  invariant. 

For  example,  let  us  have  the  algebra 


ei 

eg 

^3 

«! 

ei 

0 

0 

e^ 

0 

^2 

es 

ea 

«8 

0 

0 

Then  €„=  e^  -\-  e^ 


'SiHEFPEKS  1,  3,  3;  Caktan  3;  Hawkks  1;  SuAW  4.     Cf.  B.  Peikce  1,  3.     Fkobemus  14. 

-■  See  references  to  §  59.  3  Taber  4.  «  Tabbb  4. 


28  SYNOPSIS  OF  LINEAR  ASSOCIATIVE  ALGEBRA 

If  we  put  ^  =  ej  +  Cg,  we  find 

^  {ei  +  e3)  =  {e,  +  e,)  ?  -  e„  =  0  ^  ^3  =  0 

hence  the  characteristic  equation  (^  —  Cq)  ^  =  0,  and  by  §§61,  62, 
xi  =  ^  X2  =  e2  —  e-i  eo=  xi  +  x„ 

These  determine  the  same  algebra  (in  the  sense  of  invariant  equivalence) 


Xi 

Xs 

^3 

^I 

Xi 

0 

0 

J£2 

0 

Xo 

^3 

^3 

«3 

0 

0 

and  the  partial  moduli  are  not  the  same  as  before,  being  e^,  e^  in  one  case, 
Cj  -1-  eg,  Co  —  Cg  in  the  other. 

64.  Theorem.  If  ^,-  is  any  number  in  the  i-th  pre-  (post-)  region  of  ^,  and 
if  a  is  any  number  of  the  algebra,  then  ^j  cr  (cr  ^,)  is  a  number  wholly  in  the 
i-th  pre-  (post-)  region.'  Consequently  the  numbers  in  the  i-th  pre-  (post-) 
region  form  a  sub-algebra. 

65.  Theorem.  Let  the  numbers  defining  the  i-th  post-latent  region  of  ^  be 
^ii,  where 

y  =  1 Ci  5=1 Wji  f=  I [^^J 

We  have  of  course 

S8f  •  ^i  —  Cst  +  l 

so  that 

Then  by  §  64  the  product  of  any  number  a  gives 

"    •   Ssl     •^    •    "-Mt^    hUV 

Hence  when  these  coefficients  a  are  known  we  know  the  product  of  a  into 
any  number  of  the  form  ^H,  for 

where"  Euv^i^i  must  be  zero  if  v  +  f —  1  >/u,a. 

66.  Theorem.  If  r  is  any  number  of  the  algebra  which  satisfies  the  equation 
T.  6/  =  0,  where  r .  6,'^  '  4^  0;  then  r  must  be  in  the  region  (§  13)  2'^,  and 
in  no  lower  region.'' 

67.  Theorem.  If  r  is  any  number  of  the  algebra,  and  if  cr,^.  lies  in  the 
region  i'/.,  hut  in  no  lower  region,  then  tct,,  lies  at  most  in  the  region  2'/g, 
and  may  lie  wholly  in  lower  regions.* 

I  SuAW  4.  «8haw4.  »Siiaw  4.  ■>SiiAw4. 


TllK  ('IIAI{A(n'EUISTIC  EQUATION  (JK  A  NUMBER  29 

68.  Theorem.  Let  i''  ho  the  region  to  which  .  0'"ti~''  reduces  the  whole 
i-tli  post-latent  region,  and  generally  i'"  be  the  region  to  which  .  fl'^a""'  reduces 
the  latent  region.  Then  if  t  is  any  number  of  the  algebra,  and  cr*"  any  number 
of  the  region  i'",  then 

ra"  =  'a'",  a  number  of  the  region  1.'"  or  lower  regions.' 

69.  Theorem.  If  oJ|  is  a  number  common  to  both  regions  2'*  and  2,(,  then 
T  .  aJt  =  'a'tl,  a  number  in  the  same  regions.^ 


70.    Theorem.    Let 


Sl^.  be  the  region  \^i\\,  (s  =  1  .  . . .  w^) 

then  St  =  S'^..__a  II  belongs  to  the  regions  2''"^-''  +  '  and  2.-,„  ._„  +  ,.    Then  if  t 

is  any  number,  t  .  «S''i  =  |  S't!,  ■  ■  ■  •  S'l,ul   .  .  .  .\   for  all  values  of  t    subject  to 

the  conditions 

a«>  I  a  5<"  <  6  a<'>  +  6<"  =  ^,;,  +  1 

This  may  also  be  expressed  in  the  following  statement : 

where  y<  flu,  and  CJ;  belongs  to  S^^.^^y^^,  and  c^j  belongs  to  Sl.._t  +  i. 

Hence 

2/  ^  «  fiik  —  yt  fiy  —  t 

that  is 

y  =  t  +  ^ik  —  fiij 
Or  finally,^  if  jm^^.  <  fA^j,  then  i«,^.  i  ?/  >  < 

if  /"a>i"u,  then  ^a>  ?/  =  i +fiik  — f^ij 

It  is  to  be  remembered  also  that 

It  is  evident  that  the  products  into  £?,i  determine  all  the  other  products. 

71.  Theorem.  Since  the  units  of  the  algebra  may  be  the  numbers  £■**,  as 
these  are  mutually  independent  and  r  in  number,  it  follows  that  among  the  iv^ 
constants  of  the  algebra,  y,  which  the  coefficients  a  reduce  to  in  this  case,  there 
are  many  which  vanish  and  many  which  are  equal.  The  units  may  be  so 
chosen  in  any  algebra  that  the  corresponding  constants  y  become  subject  to 
the  equations  for  the  coefficients  a  in  §  70  [but  this  choice  may  introduce 
irrational  transformations]. 

'Shaw  4.  -^ShawM.  ^Soaw  4. 


30  SYNOPSIS  OF  LINEAR  ASSOCIATIVE  ALGEBRA 

72.  Theorem.  Since  the  iderapotents  for  ^,  viz.,  xj,  X2,  •  •  •  •  Xj„  may  be 
used  as  pre-multipliers  as  well  as  post-multipliers,  the  units  ^ij,  and  therefore 
all  units,  may  be  separated  into  parts  according  to  the  products 

X,  .  Hi  (a  =  1   .  .  .  .  iO 

As  these  parts  are  linearly  independent,  and  as  the  z-th  region  is  defined 
already  by  the  units  p^i,  it  follows  that  the  independent  units  derived  by  this 
pre-multiplication  must  also  define  the  region,  and  as  the  shear  regions  were 
unique,  their  number  for  each  shear  remains  the  same  as  before.  We  may 
use  a  new  notation,  then,  indicating  the  pre-  as  well  as  the  post-character  of  ^, 
and  at  the  same  time  uniting y  and  s  into  a  single  subscript,  thus  the  units  are 

^V        \u't'  hn"t"j  ^i" 

where 

U"—  10^,,  +    .  .  .       +    Wj„  i„  +  S  U'  =  U\,,  +    .  .  .  .    +Wj„i„  +  s 

73.  Theorem.    Let  us  return  to  the  equation  in  §7  0,  in  the  new  notation, 

'■    •      SMt    —  ""xy      <^x?y 

y-fixlyH  y=t  +  fifix  —  l-hu  a'/y  =a'fy-l=  — «it-t  +  i 

If  r  is  confined  to  expressions  belonging  to  the  region  \^ll\,  then  letting  ry 
be  any  such  number, 

—^aa      a  y  a        V"  j^aa   a  t^a  l^     ^  ^  aa    a  Ka 

^  1    •     Kul   ^  "arl      i^xl   "T    -^  "-xl      Kxy 

If  we  let 

then 

-Tj" .  CTj"  =  2  al\  zl"  °^^i  -f  terms  for  which  y  >  1 

Hence  if  we  let  tJ°  be  in  turn  each  unit  °^°  in  this  region,  we  shall  find 
from  al"  by  the  process  used  in  the  beginning  of  the  problem,  certain  numbers 
idempotent  so  far  as  this  region  is  concerned,  and  which  will  be  linearly 
expressible  in  terms  of  °^°i.  These  new  x'?.  are  linearly  independent  and 
commutable  with  x„,  since,  if  x'^  is  one  of  them,  x^xl==- x'^^=  x'^x^.  Hence 
x„  must  be  the  sum  of  them.  We  might  therefore  have  chosen  for  ^  a  number 
which  would  have  had  these  idempotents,  and  we  may  suppose  that  the 
number  'C,  has  been  so  chosen  that  no  farther  subdivision  of  the  idempotents 
is  possible.^ 

74.  Theorem.  It  is  evident  tliat,  as  the  expressions  in  the  i-th  latent  region 
of  ^  form  a  sul)-a]gebiii,  we  may  choose  one  of  them  ^[  just  as  we  clioose  |, 

'Cf.  MoLiBN  1. 


THE  CHARACTERISTIC  EQUATION  OF  A  NUMBER  3 1 

and  using  it  as  a  post-multiplier,  divide  this  i-th  latent  region  itself  into  sub- 
regions  corresponding  to  the  latent  regions  of  ^1  in  I^J.  Each  such  sub-region 
becomes  a  sub-algebra.  We  may  evidently  so  proceed  subdividing  the  whole 
algebra  into  sub-regions  until  ultimately  no  sub-region  contains  any  number 
which  used  as  pre-multipiier  has  more  than  one  root  for  that  sub-region. 
This  root  may  then  be  taken  as  zero  or  unity.  If  then  the  sub-region  be 
represented  by  cr, ,  a.,  .  .  ■  ■  a,.,,  we  have  for  every  number 

a  =  ^xj<rj  T  =  l  f/j  Gj  {j=  I  ....  /) 

T  a  =:  ^  T  -f-  t'  t/  a  =  (/ r'  -\-  r" 


Hence  if  x,  is  the  partial  modulus  for  this  region  delined  by 

o'  -  hga"^'  +  ^'-^(*~  ^^  fa"-''  ....  +  (-)"'  y"  i  a 

^.= ^ 7 

we  nmst  have  a  =  g  xi  +  ^^^  -\-  other  terms  whose  post-product  by  r  is  zero. 

Multiplying  every  number  then  by  x^ .  ()  we  arrive  at  a  sub-sub-region 
which  gives  a  sul)-algebra  whose  modulus  is  x^,  and  such  that  if  a  is  its 
character,  every  number  in  it  has  the  character 

(ua) 

This  algebra  is  a  Peirce  algebra.  Its  structure  will  be  studied  later.  The 
Peirob  algebra  is  the  ultimate  subdivision  by  this  method  of  the  algebra  in 
general  and  its  structure  really  determines  the  main  features  of  the  structure 
of  the  general  algebra. 

75.  An  algebra  may  contain  an  infinity  of  units,  in  which  case  it  may 
not  have  an  equation  at  all.     Thus  the  algebra  may  have  fur  units 


oo 


so  that  p  =  i  a-j  e^ 


!=0 


It  may  very  well  happen  then  that  pa  =  t<y  has  no  solution.     The  theory 
of  such  algebras  will  be  developed  in  a  later  paper. 

76.   Theorem.    Let  the  general  equation  of  a  number  ^  be 
Let  us  put  ^- — ???!  ^-|-(T  =  0.     Tiien  we  may  eliminate  ^  from  these  two 


32 


SYNOPSIS  OF  LINEAR  ASSOCIATIVE  ALGEBRA 


equations,  by  using* determinants,  arriving  at  an  equation  in  terms  of  a  of 
order  r.     Thus  we  have 


1 

1 

0 


+  "'s 


ni, 


m. 


0 
a 


(-Ifw,  0 

0  0 

0  0 

0  0 


0 
0 


0 
0 


0 
0 


m. 


0 


=  0 


or 


1 

—  WJl 

+  nio      .  .  •  • 

0 

1 

— wi      .... 

0 

0 

a  —  m„      .  •  •  ■ 

a  —  ?»2 

OTg 

—  m^      

0 

O — tOo 

WJg             

1 

—  ?Hl 

(T          .  .  .  . 

0 

1 

—  nil      .... 

The  highest  powers  of  this  equation  are 


a"—  2771,  CT*— ^  + =0 


Hence  the  sum  of  the  roots  of  0  is  2w2. 

77.  Theorem.  In  general  if  ^*  —  Wi  ^*""^  +  ....  +  ( —  1)'  a^  =  0,  we  find 
in  the  same  manner  a  determinant  of  order  r  -\-  s,  reducing  to  one  of  order 
r  in  (Tj,  the  first  two  terms  becoming 


,r_l  + 


0 


Hence  for  any  such  number 

we  have  the  sura  of  the  roots  of  a,  equal  to  stti^.    Hence  the  s-th  scalar  coefficient 
m,  of  ^  is  i  into  the  scalar  coeflBcient  of  order  unity  of  {—Y^^^x^'^  (^)  >  '^^ 

78.  Theorem.    We  may  also  find  the  general  equations  of  tlie  cr's,  and  in  a 
similar  way  of  the  ;^'s. 

79.  Theorem.    In  this  way  one  may  form  the  equations  of  powers  of  ^,  or 
of  any  polynomial  in  ^. 

80.  Theorem.    Let  there  be  formed  for  any  number  p,  the  products 

pe,  (1=1....  ;•) 

The  e<  form  the  basis  and  are  orthogonal.     Then  we  have  (p  —  (j)  a  =  0,  when 

V   .  I  .e,  (pp,)  I .  ejG  =  (J  I  .  c'iG  {i  =  l  ....  r) 


TIIK  CHAUACTKIIISTIC   KtiUATK^N  OK  A  NlIMriEH 


33 


Hence 


/.  e,  ((X',)  — J/       /.e,  (j;e.,) 


=  0 


or 


gr_fjr-i   V  ,  / .  e,  (pC,)  +  f      ■'    ^      I  .C^   ACj  ^(pC,)    (pCj) 


=  0 


i-  1 


lj=l 


This,  however,  must  correHpoiid  to  the  general  pre-latent  equation  of  p,  and 
therefore 


Wg  =    ^   .  /.  Cj  ^e;  -4(pe,)  (pey) etc. 


Thus 
Therefore 


81.    Theorem.    We  have  at  once 

r 

j(;  .a  =  {m[  —  p)  .  (T  =  2  .  (ct  /  .  e,  (pe,)  —  (pe,)  /.  e.a) 

r 

(=1 


2 !  Hi^  =  2  .  /.  c/,  (p  .  j^'^a)  =  2/ .  e,,  (pe^)  / .  e^  Ae^  Ae^  (pej 

A: 

=  2  .  /.  (pe^)  e,^.  /.  e^.  ^(pe,)  ^e^  e^ 
=  2  .  / .  (pe,)   ^(pej)  4e,.  e^ 
=  2  ./.  ei^e,.  ^(pej)  (pe,.) 

Since  ;^" .  ct  =  (wig  —  p  .  'x!)'^>  ^^^  have 

;jr".  (y=-Z  A  .e-.ejAa  (pe^  (pe^) 
In  general,  we  find 

X^'\  a  =  S  .Aeiej....e,  Ja(pe,.)  (pe,)  ....  (pej 

82.    Theorem.    If  we  use  the  notation  of  the  ^-pairs,  these  become 
m[  =  /^pO 


»j 


's  =  --,I.^,A^,....C.AipQ{pC^....ipQ 


;K<»'.ct  =  -,  J^,....^,^(T(p^,)....(pQ 

In  this  form,  the  independence  of  the  expressions  m  and  x  from  any  particular 

unit-system  is  shown. 

83.    Theorem.    Let  us  write  further 

m'  (p, ,    p, .  .  .  .  p,)  =  -J  7^1  A^,....  ^,^(p,  4^,)  (p„  Q  ....  (p_,  Q 
Then,  from  the  properties  of  the  ^'s,  this  form  will  reduce  to 


»''(pi,  •  ■  •  • ,  pj  =  „,    [»'i  (pi)  ^"i  (p^)  •  •  •  ■  m^  (pj 


2  .  wi;  (p,)  (w{  (pg)  ....  m[  (p,    1  pj  +  .  .  .  .  ] 


34  SYNOPSIS  OF  LINEAR  ASSOCIATIVE  ALGEBRA 

according  to  the  rule :  Insert  m'l  before  every  selection  of  p's  taken  according 
to  the  partitions  of  s,  giving  each  term  of  s  —  Jt,  factors  the  sign  (— )",  and 
writing  in  each  factor  the  product  of  the  p's  in  every  order  possible  when  the 
p  of  lowest  subscript  is  kept  first  in  the  product.  For  example,  if  s  =  3,  we 
have  the  partitions  3  =  1  +  1  +  1  =  1  +  2—3.     Hence, 

7w'(pi,  p.,,  Pa)  =  Yl  t"*!  (Pi)  ^'^  (P-')  m'l  (ps) 

—  m[  (pi)  "»i  (p2  p:))  —  '"i'  (pa)  ^i  (pi  Ps)  —  "'i  (p.3)  '"1  (pi P2) 

+  n'l'  (pi  P2  p3  +  Pi  p.3  pu)] 
We  note  that 

m'  (pa  p/, p,  p/)  =  »"'  (p/  pn  p;, pc) 

If  s  =:  r  +  1,  this  form  must  vanish  identically. 

84.  Theorem.    If  we  put 

j(^{p,....p,)a  =  ^^^A.^,....^,Aa  (p,  ^i)  (p.  ^2)  ■  •  •  •  (ps  Q 
then,  if  Xi  stands  for  [x  (pi)]  (o),    X\'i  ^o'"  \x  (pi  P^)]  <^»    ^t<^-> 

;t(pi  ••••  p.)<^  =  ^,|[%i-;t2  ••■•  r^(<^)  — 2.%i.%.3  ••••  3:s^i,.(<t)  +  ----] 
The  rule  is  the  same  as  for  the  preceding  expression  of  m\  thus 

X  (pi,  p3,  Pa)  •  <J  =  ^,  [;ci .  ;t3  •  Iz  (ff) 

—  ;i:i  a:-'3  (<t)  —  Z3 ;i:i3  (<7)  -  xz xvz  {<^)  +  {x\zz  +  ;i:i33)  <y] 

85.  Theorem.  If  ???,,,  ,,,...., ,,  is  the  function  2  .  y\'  g-i  ■  ■  •  -gl' ,  summation 
over  all  permutations  of  1,  2  ....  t,  then 

"»..,  ......  =  /^'i^iV  ■  •  •?. ^ (p-^-  ?i)  (p^^  ^.)--  ■■{f'^t) 

86.  These  numbers  m  and  functions  ;^  are  called  invariants  of  p,  or  of 
Pj,  p.j  .  .  .  .,  as  the  case  may  be,  since  they  do  not  depend  on  any  particular 
system  of  units.  It  is  obvious  that  any  function  of  pi,  pg  .  •  •  ■  pt,  containing 
only  ^-pairs,  is  an  invariant^  in  this  sense. 

87.  Theorem.    If  p  a  =  0,  then  x'  ■  u:- =  »'i  «,  p  ;|^' .  a  =  w?,  p  a  =  0 

x"  .  a  ^  Wo  "- 
^'**  .a  =■  m^ .  a 
In  general,  if  p  a  =  <;  a,  then 

j^.a  =  {mi  —  g)a %'"'  .  a  —  (w,  —  m,^^  g  -\- ±(7")  a 

If  pay  =  g  uj  +  ao         p  a^  =  gf  a, 

;^w  .  a,  =  (m,  —  w«_i  g  +  ....±g')a,  —  {m,_,  —  m,_.,  g....  Tj/""')  a^ 
Similar  results  may  be  found  for  the  other  latent  regions  of  p. 


I  ct.  id'AuLAT  1. 


THE  CHAUACTERI8TIC  EQUATIONS  OP  THE  ALGEBRA  35 

m.     THE  CHARACTERISTIC  EQUATIONS  OF  THE  ALGEBRA. 

88.  Theorem.  Of  Hh!  miita  taken  to  define  the  alj^ebra  in  the  preceding 
chiii)ter,  certain  ones  will  Ijo  of  pre-character  a,  po.st-cliaracter  /?.  Let  the 
number  of  such  be  represented  by  r/,,^ .  Then  the  total  nuinljcr  of  those  of 
post-cliaracter  ji  will  be 

It'll  =7lifl  +  Ihn   + +   S^ 

The  number  of  pre-character  a  will  be 

nl  =  «ai  +  «a3  + +  n^v 

89.  Theorem.  We  may  state  the  general  multiplication  theorem  again  in 
the  following  form,  ^  being  any  number : 

V 

where 

k'  —  kt  0  (iiy  —  /i  >  k'  —  k  X  ftij,  —  |U„j 

In  this  equation  each  coefficient  o  is  a  linear  homogeneous  function  of  certain  of 

the  coordinates  a-  of  ^,  namely  those  of  type  a-',;,"*  where  '^u,,  combines  with 
°^^/,  without  vanishing. 

90.  Theorem.  If  we  multiply  ^  into  each  unit,  and  form  the  equations 
resulting  from  the  pre-latent  equation^  of  ^,  say  A' .  ^  =■  0,  we  have  at  once, 
because  the  units  have  been  chosen  for  the  post-regions  of  a  certain  number  ^, 

The  orders  of  these  determinant  factors  are  n'l,  n'o  ....  n",  their  sum 
being  equal  to  r. 

91.  Theorem.  An  examination  of  the  determinant  A,'  shows  that  it  may 
be  divided  into  blocks  by  horizontal  and  vertical  lines,  which  separate  the 
different  units  *^ii,  ^^l-^,  ....  according  to  the  power  of  6^  which   produces  the 

units,  the  order  being 

c  ,         c 

^ul  •  •  •  •  skmjj 

There  are  ^,,  columns  and  rows  of  blocks.  But,  from  the  properties  of  the 
coefficients  a,  the  constituents  in  the  first  block  on  the  diagonal  are  the  only 
constituents  in  any  block  on  the  diagonal.     Hence  we  may  write'^ 

a;  =  A;r»  A^«3 ....  A/^ic. 

92.  Theorem.  The  determinants  A-,.,  s=  1  ....  C(,  are  irreducible  in  the 
coordinates  of  ^,  so  long  as  ^  is  ant/  number.  For,  if  one  of  these  determinants 
were  reducible,  then  the  original  separation  by  idempotents  could  have  been 
pushed  farther — as  this  separation  was  assumed  to  be  ultimate  no  farther 
reduction  is  possible.^ 

'On  the  general  equaUon  see  Studt  3,  3;    Sforza  1,  2;    Scheffbbs  1,  2,  8 ;    Molien  1;    Cartan  2; 
SUAW  4;    Tabeu4;    Fkobenius  14. 

»  Shaw  4.     Cf.  Caktan  3.  « Ct.  Caktan  3. 


36  SYNOPSIS  OP  LINEAR  ASSOCIATIVE  ALGEBRA 

93.  Theorem.  Confining  the  attention  to  A'^.,  lei  the  units  °^ji  whose  pro- 
ducts by  ^  give  A«.,  be  7i  in  number,  with  the  pre-characters  a=  1,  ••••,/, 
/^  h.  The  coordinates  x  appearing  in  the  coefficients  a,  must  be  of  the  form 
x'*'"'-'.  It  follows  that  if  ^  be  chosen  so  that  all  coordinates  x  not  of  these 
characters  (uia.,),  aj,  a^^  1  ■•••/,  are  zero,  then  the  value  of  A,';^  will  not 
be  affected.  The  aggregate  of  such  numbers,  however,  obviously  constitute  a 
subalgebra  which  includes  x„,  a=:  1  ..../.  These  numbers,  say  ^''''^,  when 
multiplied  together  yield  a  pre-Iatent  equation  ^„„=  0,  which  must  be  a  power 
of  A-^.,  and  therefore  irreducible.  It  follows  that  if  we  treat  this  subalgebra 
as  we  have  the  general  case,  we  shall  find  but  one  shear  making  up  the  whole 
of  each  latent  region.     Consequently  the  units  of  this  algebra  take  the  form 

e„,a,  («!,  a2=  1 /) 

They  may  be  so  chosen  that 

The  partial  moduli  are  evidently^ 

e»,a,  (ai=l  ....  /) 

94.  Theorem.  Since  any  unit  °£^  may  be  written  e.^  °^^  it  follows  that  no 
expression  e^,,.  "^^  can  vanish,  else 

Hence  if  there  is  one  unit  °^^,  there  are  all  the  units' 

95.  Theorem.  The  units  of  the  algebra  may  therefore  be  represented  by 
the  symbols 

e«3  e^y  Ss 

where  the  numbers  e^'  and  e^j*  are  such  that 

f'a?    '^yS     <^^*:    o'^/Sv   t'ai 

The  numbers  e^^  form  an  algebra  by  themselves,  such  that  its  equation  consists 
of  linear  factors  only,^  as 

A,  =  {a^  -  0 

96.  Definition.  An  algebra  whose  equation  contains  only  linear  factors 
will  be  called  a  Scheffers  algebra.  If,  further,  it  contains  but  one  linear 
factor,  it  will  be  called  a  Peirce  algebra.  If  it  contains  fiictors  of  orders 
higher  than  unity,  it  will  be  called  a  Cartan  algebra.  An  algebra  consisting 
of  units  of  the  type  e[^l  only,  will  be  called  a  Dedekind  algebra.*  The  degree 
of  an  algebra  is  the  order  of  its  characteristic  equation  in  ^. 


'  Moi.iEN  1  (urspriingliche  systeme);    Cartan  1,  3;   Suaw  4;    Fuohenius  14. 
'  Cautan  2;    Fkobenius  14. 

^Caktan   1,  3.      On    tbe    "multiplication"    of    algebras  by  eacb   iitlier,   see   Ci.iffoud  8 ;     TAiimt   1; 
ScHEKPEii.t  ."!.     Cf.  Taiiku  4;   Hawkes  1,  2;   Fkoiienius  14. 

•On  classification  see  ScuEFKEUS  K,  4;    Moi.ikn  1,  2,  3  ;    Caiitan1,2;    Siiaw4;    B.  1'k)U(:e,  1,  .S, 


THE  CHARACTERISTIC  EQUATIONS  OF  THE  ALGEBRA  37 

97.  Theorem.  Let  the  algebra  be  of  the  Schefi'er's  type.  The  irreducible 
factors  of  it.s  pre-latent  equation  are  all  linear;  hence  in  the  latent  post-region 
of  any  root  of  ^,  the  shears  are  of  wi'llh  unity  only.  The  units  defining  ihe 
i-ih  region  become 

"^jt  a=  1  .  .  ■  .  p         j=  I  ....  Ci         1=  I  ....  /j^j 

Ihi  >  t^is >l"'Oi 

The  product  of  ^  into  any  unit  is' 

S  •    S.H  —  —  (*fjk--k     KflC 

where 

Ic'  —  hlO         fi^j,  —  /.-  >  k'  —  /.- ;:  fii,y  —  (I,  J         f  -1  j  if  Id  =  h 

98.  Theorem.  If  we  remove  from  this  algebra  all  idempotent  units,  the 
remaining  units  form  a  nilpotent  algebra  of  r — ^j  dimensions.  The  equation 
A'  ^  =  0  reduces  in  this  case  to  a  determinant  whose  constituents  on  the 
diagonal  and  to  the  right  of  the  diagonal  all  vanish,  hence  it  is  evident  that 
the  product  of  any  two  of  its  numbers  is  expressible  in  terms  of  at  most 
r — p —  1  numbers.  Let  the  original  units  be  <^,,+i,  <?»p+2  ■  •  •  •  4'r-  Then  the 
products  ^,,4),..   do  not  contain  a  certain  region  defined  by  a  set  of  units 

%^-i    ■■■■  ^p+h,  {h  >o) 

The  products  of  these  h^  units  (which  constitute  the  region  fj,  let  us  say,) 
among  tliemselves  and  with  any  other  units,  are  linearly  expressible  in  terms  of 

%+in+t  (<  =  1  •  •  •  •  r  —  p—h,) 

Similarly  any  product  <^,^  ^,.,  <^,^  can  not  contain  a  region  e^,  defined  by 

Hence  {e^.Jfal-,  |%!  ■  jfij-,  ^"^  {f^} -Ifsf  depend  only  on  <^p^t,  <  ^/fi^  A,. 
Proceeding  thus,  it  is  evident  the  domain  of  the  nilpotent  algebra  may  be 
separated  into  regions  defined  by  classes  of  units  which  give  products  of  the 
form 

\^c\  .  \ej\  =  \e,]  {k>i,  k:>j) 

In  particular,  the  units  of  the  Scheffer's  nilpotent  algebra  may  always  be 
chosen  so  that,  if  they  are  >:,,  yj^   .  .  .  .,  then 

>7i  Tj  =  2  Yijk  >7t  {k  >  J,  k  >y) 

It  is  also  evident  that  for  any  r  —  ^)  +  1  numbers  ^t  we  have 

The*  products  of  order  I  form  a  sub-algebra  of  order  r', 

r'  <  ;•  —  Z  -h  2 


'Shaw  4,  5.  -'^ScHErFERS  3  ;    Cartan  3  ;    Shaw  5  ;    Frobesius  H. 

3 


38  SYNOPSIS  OF  LINEAR  ASSOCIATIVE  ALGEBRA 

99.  Theorem.    In  any  Cartan  algebra  the  units  may  be  so  taken  as  to  be 

represented  by 

e%  {i=l  ....  p  a,(3=l....  tCi) 

yii'ii        (*,  y  =  1  —  i>  «;  i^  =  1  —  w'i) 

The  laws  of  multiplication^  are 

100.  Theorem.  Returning  to  the  Scheffers  algebra,  if  we  retain  only  its 
nilpotent  sub-algebra  and  the  modulus,  we  shall  have  a  Peirce  algebra.  The 
equation  of  this  algebra  will  contain  but  a  single  factor  and  the  pre-  and  post- 
characters  of  its  units  may  be  assumed  to  be  the  same.  The  nilpotent  6  becomes 
the  sura  of  the  nilpotents  Bi  +  do.  .  .  .  +  6^,.  The  product  of  ^  into  any  unit 
may  be  written^ 

S  •  hjk  =  2  dfjki-k  hj'k' 

k'  —  klO  ny  —  k>k'—ktiiy—Hj  j'lj\ih'  =  k 

101.  Theorem.    Let  the  characteristic  equation  of  any  number  be 

^- _  /j  .  ^'"-1  +  ....+  (-)»'/™  =  0  {m<r) 

where  f^  is  a  homogeneous  function  of  the  coordinates  of  order  i.  Differen- 
tiating this  equation,  and  remembering  that  d^  is  any  number,  we  arrive  at  m 
general  equations  connecting  1,  2,  .  .  .  . ,  m  numbers  of  the  algebra :  as 

{^f-'^.  +  ^'r'^2^.  +  •  •  •  •?2^i"-^)-  [/i  [Q^T' 
(^r-^^2^3  +  ----)-etc.  =  o 

These  equations  are  the  second,  third,  etc.  derived  equations  of  the  algebra, 
according  as  they  contain  two,  three,  etc.,  independent  numbers  ^i,  ^2> 
etc.  These  equations  lead  to  many  others  when  the  scalars  of  ^  are  intro- 
duced.^ The  new  coefficients  fi{<^a,....'(a.^  will  be  called  the  scalar  charac- 
teristic coefficients  of  order  i  for  ^„,  ....  ^„..  They  usually  differ  from  the 
coefficients  m. 

102.  Theorem.  The  general  equation  of  r  numbers  of  the  algebra  of  order  r 
is  written  (2  representing  the  sum  of  the  r!  terms  got  by  permuting  all  the 
subscripts) 


'CAKTAK8.  »8haw4,  6.  sTabbk,  2,  3.     Shaw  4. 


TUK  CHAKACTKUI8TIC  EQUATIONS  OF  THE  ALOEBUA 


39 


In  this  equation,  omitting  the  Kubscript  1,  so  that  m  =  mi 
^i  (^t,  Q       =m'(,  .  m'(j  —  m ^,  ^j  =  m,  {'(j,  Q 
"t3(^(,  (j,  </.)  =  "J^i  •  '"^J  •  '»^*  —  "»Ci  •  ^^j  '(k  —  mi^i  .  m^i^^fc 

=  w'a  (?(,  ?*,  ^j)  =  "'3  (  ^^>  ^*i  Ci) 
These  formuIaD  Ibllow  from  the  identities 

«'"«(<?>  I,  <?>1  ••  •^l)=»'l  ['"»-!  (4*1  ;<?'l  ••■<?'l)  ■1>l  —  ^^-Al>l  ■■■1>l)-1>l 

+  ...  +  (-  !)"->  »n.  («?.,  <?>,) .  <i>r-  +  (-- 1)"  '"i  <?>. .  rr'  +  (— 1)"+' .  <?>;] 

and 

w,  (<?)i  ...<?>,:...<?»,...<?).)  =  m.  (<?)i  ...<?),•..<?)(••■  <?>»)  i,  y  =  1  .  . .  » 

We  arrive  at  the  formulae  directly  by  differentiating 


s!  m,{^i,  <^, <^,)  = 


m(p^ 

1 

0 

m<pi 

m^f 

2 

mfl 

m^i 

m( 

m^r' 

m^r' 

m 

w^; 

mrr' 

m 

h»-3 


0 

0 

0 

0 

3 

0 

0 
0 
0 


TO<^, 


103.  Theorem.  A  study  of  the  structure  of  all  algebras  of  the  Scheffers 
type  gives  u.s  the  structure  of  all  algebras  of  the  Cartan  type,  as  we  may  pro- 
duce any  Cartan  algebra  by  substituting  for  each  partial  modulus  of  the 
Scheffers  type  a  quadrate,  and  then  substitute  for  each  unit  of  the  algebra  a 
sub-algebra  consisting  of  the  product  of  this  unit  by  the  two  quadrates  which 
correspond  to  its  characters.* 

104.  Theorem.  Each  Scheffers  algebra  may  be  deduced  from  a  Peirce 
algebra  by  breaking  the  modulus  up  into  partial  moduli,  accompanied  by 
corresponding  separations  of  the  units.  For,  if  all  partial  moduli  of  a  Scheffers 
algebra  are  deleted  from  the  algebra,  leaving  only  the  modulus,  and  a  set  of 
nilpotent  units,  we  have  a  Peirce  algebra.  Any  Peirce  algebra  may  be  con- 
sidered to  have  been  produced  in  this  manner,  so  that  to  any  Scheffers  algebra 
corresponds  a  Peirce  algebra,  and  to  any  Peirce  algebra  correspond  a  number 
of  Scheffers  algebras. 


105.    Theorem. 


If  the  characteristic  function  of  an  algebra  be 


AJ> 


..a;;i.=  o 


wherein  A,  is  a  determinant  in  which  ^,  the  general  number  of  the  algebra, 
occurs  only  on  the  diagonal,  and  the  other  constituents  are  linear  homogeneous 
functions  of  the  coordinates  oft,',  and  if  we  substitute  for  ^  where  it  occurs  4-1 


'Cartan  3.     Cf.  Molien  1;  Shaw  4. 


40 


SYNOPSIS  OF  LINEAR  ASSOCIATIVE  ALGEBRA 


any  arbitrary  number  of  the  algebra,  then  the  resulting  expression  may  be 
written  0(4')  =  Alj''  (4-)  AS=  {4')  ■  ■  ■  •  ^pp{'^)-    This  expression  will  vanish  only  for 

wherein  K^(^i  has  the  meaning  given  in  part  II,  chapter  XIX,  art.  3. 
Thus  the  algebra  whose  characteristic  equation  is 


(^0^0 


•'oo 

^10 


0-0  =  0 


gives  the  expression 

^u-4 
This  expression  vanishes  when  and  only  when  -^=-21,  Kq\,  or  q^;  wherein 

qz  ^^  ^a  ^^33 

That  is,  the  expression  is  factorable  into  (1^  —  q^)  {'^  —  /ij,)  (1^ —  q.?j. 
As  a  corollary,  the  expression 


«oo  — 

«10 

Q 

OqI                      •  •  •  -^On-l 

«n  —  6               «i  n-i 

«n-10 

a„_ii       a„_i  „_i  —  6 

IS 


factorable  in  the  matric  range  ofq^,  Kq^ .... /l"  ' (71. 


rV.     ASSOCIATIVE  UNITS. 

106.    Definition.    The  multiplication  formula  in  §  100  may  be  used  to  intro- 
duce certain  useful  new  conceptions.     It  reads 


Jd  —  hio 


Ki 


h>k'  ~  k^ixj,~fij 


j'  <  J  if  k'  =  k 


Let  us  consider  an  algebra  made  up  of  units  which   will  be  called  associative 
units,  represented  by  /l„(,  such  that 


where 


'k'     Citjj,      .      t    j  /;!  _|.    ^. 


k^O  i  >y  if  /.-  =  0 


c=  1  if  (Uj  >  A;  ^  jUj  —  fij, 
c  =  0  if  ju,  <  /c  <  ^/j  —  fij, 

Since  there  is  a  modulus  Cq,  and  since  £«-,  =  ^/k'Co,  every  unit  ^j,^,  is  expressible 
as  a  sum  of  these  units  2,^,;  multij^Hed  by  proper  coefficients,  and  every  number 
'(  is  expressible  as  a  sum  of  the  units  with  proper  coefficients.  Hence,  we  may 
express  ^  in  the  form 


l^i    >  ^'^    "    /<, 


k^  0         i  ]>y  when  /• 


ASSOCIATIVE    UNITS  41 

The  Peirce  algebra  i.s  expressible  therefore  as  a  sub-algebra  of  the 
algebra  of  the  af^sociative  units  whose  laws  of  multiplication  '  are 

where 

Uj  >■  /c  ^  jU(  —  jUj  Jc  :  0         i  >-y  when  7c:=  0 

fXi  >  //  ^  ^i  —  fi),         k'^  0         i  >/  when  lc'=  0 

c=  1    if  ^i>h+  k>  l^i—  Hj,      k  +  /^'  =  0      /■  >/  when  k  +/.-'=  0 
c  =  0  if  /i(  :;,  /.;  +  k'<^^i —  ^j, 

107.  Definition.  An  expression  of  an  algebra  in  terms  of  associative  units 
will  be  called  a  canonical  expression.  In  many  cases  the  associative  units  are 
the  units  of  the  algebra,  in  part  at  least,  but  the  units  of  the  algebra  will 
frequently  occur  as  irreducible  sums  of  these  units  with  certain  parametric 
coefficients.  This  theorem  extends  C.  S.  Peirce's  theorem  that  every  linear 
associative  algebra  is  a  sub-algebra  of  a  quadrate"  of  order  r^. 

108.  Theorem.  The  Schelfers  algebras  derived  from  this  Peirce  algebra 
have  partial  moduli  of  the  form 

«t  =  ^hho  Oi='^  ■■■■  Oi 

When  each  partial  modulus  ei  is  of  the  form  ?.jio,  the  SchefTers  algebra  coincides 
with  the  algebra  of  which  the  Peirce  algebra  is  a  sub-algebra.  Such  SchefTers 
algebras  will  be  called  primary  algebras.  The  units  in  any  Scheffers  algebra 
are  separable  into  classes  according  to  their  characters,  those  of  chai'acter  j 
having  in  their  expression  units  X  of  the  type 

7.,j.„  or  %j.,k  j\=l (Ji 

109.  Definition.  The  units  of  a  SchefTers  algebra  are  separable  into  those 
of  characters,  (aa),  and  those  of  characters  (a/3),  a  :^  ^S.  Those  of  characters 
(aa)  constitute  the  direct  units.     Those  of  characters  (a/3)  are  the  skew  units.^ 

110.  Theorem.  The  pre-latent  (post-latent)  equation  must  contain  the 
factor  (aj.o  — ^)  to  that  power  which  is  the  sum  of  the  multiplicities  belonging 
to  i: 

The  characteristic  equation  will  contain  {a^Q — ^)  to  that  power  which  equals 
the  maximum  multiplicity^  fif\ 

111.  Theorem.    A  Cartan  algebra  will  have  for  a  canonical  expression 

^  ^  2  a^k      ?'-a/30  ^ijk 

where  the  units  Z  and  ?J  are  independent  of  each  other. 

I  Shaw  4,  5.       sc.S.  Peikce  1,  4.       'Schkffers  3.       <  Scheffers  3;  Shaw  4. 


42  SYNOPSIS  OF  LINEAR  ASSOCIATIVE  ALGEBRA 

112.  Theorem.  We  may  obviously  combine  these  forms  into  still  more 
compound  expressions  as 

^  =  2  a(ij,ki)iuhh)  •  •  ■  •  ■^'id,k,  '^"uj,k,  ■  ■  ■  ■ 
Such  numbers  are  evidently  associative,  and  could  be  considered  to  be  the 
symbolic  product  of  algebras  with  only  one  2.. 

113.  Theorem.  Returning  to  the  equations  of  the  algebra  §108,  we  see 
they  evidently  depend  on  those  associative  units  which  are  of  weight  zero. 
The  equations  are 

characteristic :     0  =  A  .  ^  =  Aj"  AJ^ ....  A^pi 

p 

pre-latent :     0  =  A'.  ^  =  11  .  A;  ^'=' 

i  =  l 

P 

post-latent:^     0  =  A".  ^=  11  .  Aj^=i  ^' 

1=1 

114.  Theorem.  The  number  Aj  (^)  can  not  contain  any  associative  unit  of 
the  form  /ljj,o>  where  the  constituents  of  Aj  are  of  the  form-  ajj,o> 
/j  =  1  ....</!.     The  factor  A,  (^)  is  the  i-th  shear  factor  off. 

115.  Theorem.  The  product  Aj  f  .  A3  if  can  xiot  contain  any  associative 
unit  of  the  form  X^^j, p;  or  X;„^,,,.  The  theorem  may  be  extended  to  the 
product  of  any  number  of  shear  factors.- 

116.  Theorem.  The  product  (Aj  f)'"  can  not  contain  any  associative  unit 
of  the  forms 

'^j'liiOj  \y,  1  •  •  •  •  X;,j-,  m_i 

117.  Theorem.  The  third  subscript  in  X^^^,  h,  is  called  the  weight  of  X. 
Every  number  f  may  be  written  in  the  form 

f  =  f  (»)  +  f  (»)  +  ••••  +  f  <""  a  i  0,  i  >  a 

The  weight  of  f  is  the  weight  a  of  its  lowest  term.     The  weight  of  the  product 
of  two  numbers  is  the  sum  of  their  weights. 

118.  Theorem.  The  terms  f**  constitute  an  algebra.  This  may  be  called 
a  compa7iion  algebra,  and  may  or  may  not  be  a  sub-algebra  of  the  given 
algebra.^  The  quadrate  units  of  an  algebra  evidently  belong  also  to  the 
companion  algebra. 

119.  Theorem.  To  every  transformation  of  the  units  of  a  companion  alge- 
bra corresponds  a  transformation  of  the  units  of  the  given  algebra.     Hence 

'Cartan2.  'Shaw  4. 

^  Cf.  MuLiEN  1.     "Begleitende"  systeniB  include  these  companion  algebras,  and  may  or  may  not  be  sub- 
algebras  of  the  given  algebra. 


ASSOCIATIVE    UNITS 


43 


the  ^*'"  terms  may  jilwayH  be  taken  according  to  the  simplest  form  for  the 
companion  algebra,' 

120.  Theorem.    If  the  general  equation  of  an  algebra  is 

'C  —  Wi  ;"■  '  +  Wg  ^'•-2 =0 

a 

and  if  when  ^  =  2  a-jC?,  we  put  y  =  S  .  e,-^ —  ,  then  v  •  ^a  =  0  gives  r  erjua- 

tion.s,  not  necessarily  independent,  from  which  the  r  coordinates  may  be 
expressed  linearly  in  terms  of  rj  arbitrary  numbers.  These  determine  the 
nilpotent  system;  or  from  the  r — Ti  coordinates  which  vanish,  the  Dedekind 
sub-algebra." 

121.  Theorem.    Since  v  =  ^  -^^  V,  and  /^  v  •  p  =  ^>  therefore 

V . «',  (p)  =  V  •  H.  (p^i)  =  ?. .  /^2  V  •  /^i  (pCi)  =  ^2  /^i  (^^i) 

But  /.  i,'i(^a^i)  =  nii{(^.,),  therefore  we  have 

V  J"!  (p)  =  ^2  "ii  {Q 
mi  (e<)  =  0  t  =  1  .  .  .  .  r 

V  m,  (p)  =  ^3  /^i  A^,  A  [(^3  ^,)  (pC^)  -  (^3  ?2)  (p^'i)] 

=  2^3 


This  can  vanish  only  if 
Again, 

hence 


=  2  :£  e 


/^2(^3^l)    /^3(P?2) 


i  =  l 

r 

=  226;  [wii  (Ci) .  ?Hi  (p)  —  mi .  (e;  p)] 
This  vanishes  if,  and  only  if, 


or 


2  a;^  ]  mi  (e,)  m,  (cj)  —  m^  (e,  Cj) }  =  0 


t  =  1 


t=  1 


These  are  the  equations  referred  to  in  §120.     The  method  used  here  has  an 
obvious  extension. 


'  Of.  Shaw  5. 


'Cartas  2. 


44  SYNOPSIS  OP  LINEAR  ASSOCIATIVE  ALGEBRA 

V.    SUB-ALGEBRAS.    REDUCIBILITY.    DELETION. 

122.  Definition.  A  sub-algebra  consists  of  the  totality  of  numbers  ^  such 
that 

^  =  Xxf  Ci  i  =  1  .  .  . .  ?•',  r'<^  r 

for  which^ 

?i  ^2  =  2  .  x[x'!  y^k  Ck  i,  j,1c  =  \ r' 

123.  Theorem.  In  a  SchefFers  algebra  all  units  with  like  pre-  and  post- 
character  (aa)  define  a  Peirce  sub-algebra.^ 

124.  Theorem.  The  Peirce  sub-algebras  formed  according  to  §  1 23  define 
together  the  direct  sub-algebra.  The  characteristic  equation  of  this  sub-algebra 
does  not  differ  from  the  equation  of  the  algebra.^ 

125.  Theorem.  The  quadrates  form  a  sub-algebra,  the  semi-simple  system 
of  Cartan/  called  a  Dedekind  algebra.^ 

126.  Theorem.  All  units  in  a  Cartan  algebra  with  characters  chosen  from 
a  single  quadrate  form  a  sub-algebra,  the  product  of  the  quadrate  by  a  Peirce 
algebra.     Its  equation  has  but  one  shear  factor. 

127.  Theorem.  All  sub-algebras  of  §126  determined  by  the  different 
quadrates  form  the  direct  quadrate  sub-algebra.  Its  equation  does  not  differ 
from  that  of  the  algebra. 

128.  Theorem.  All  numbers  which  do  not  contain  quadrate  units  form  a 
sub-algebra  called  the  nil-algebra  (Cartan's  pseudo-nul  invariant  system).* 
The  units  of  this  system  are  determinable  to  a  certain  extent  (viz.  those  which 
also  belong  to  the  direct  sub-algebra  of  §  1 27)  from  the  equation  of  the 
algebra.  The  other  units  are  not  determinable  from  the  characteristic 
equation  of  the  algebra.^ 

129.  Definitions.    All  numbers  ^,  which  are  expressible  in  the  form 

r' 
i  =  l 

form  a  complex.     The  entire  complex  may  be  denoted  by  E^,  E.,,  etc.,  E=E(, 
denoting  the  original  algebra." 

The  product  of  two  complexes  consists  of  the  complex  defined  by  the 
products  of  all  the  units  defining  E^  into  the  units  defining  E^,  indicated''  by 

E,  .  E.-, 

An  algebra  E  is  reducible  when  its  numbers  may  all  be  written  in  the 

'  On  the  general  subject  see  Study  1,  2,  3 ;    Scheffebs  1,  2,  3,  4,  7 ;    B.  Pbiboe  1,  3  ;    IIawkes  1.  2. 
'ScHEFFEKsS.     Caktan  2.  3  Shaw  4.  ■»  Cahtan  2.  i' Fkodknius  14. 

•  Epstken  and  Weddehburn  2.  '  Epsteen  and  Weddbrburn  2  ;    Fkobenius  11. 


SUB-ALGEBRAS.     REDUCIBILITY.     DELETION  45 

form  C  =  ^'i  +  C:  where  ^|  belongs  to  a  comiilox  E^,  ^.;  to  a  complex  E.,,  such 
that,' 

E,.E^=  A',  A',  .  E.,  =  0  E^.  Ei=zO  E.,.E.,  =  E., 

An  algebra  is  irreducible  when  it  can  not  be  broken  up  in  this  way. 
When  reduciljle  into  a  complexes  we  may  write 

E=E,  +  E,+  ....  +E^ 

130.  Theorem.  An  algebra  is  reducible  into  irreducible  sub-algebras  in 
only  one  way.^ 

131.  Theorem.  The  necessary  and  sufficient  condition  of  reducibility  is 
the  presence  of  A  numbers  e^  ....  e,,,  such  that  if  ^  is  any  number,'^ 

^e^  =  eX  e^  =  e.  e.  e^  =  e^  e,  =  0  a  =  1 h,  a^^ 

132.  Theorem.  The  characteristic  function  of  a  reducible  algebra  is  the 
product  of  the  characteristic  functions  of  its  irreducible  sub-algebras."  The 
order  is  the  sum  of  the  orders  of  the  sub-algebras,  and  the  degree  is  the  sum 
of  the  degrees  of  the  sub-algebras. 

133.  Definitions.  The  region  common  to  two  regions,  or  the  complex 
common  to  two  complexes  Ei,  E.,,  is  designated  by  E^.,.  If  the  complex  E^  is 
included  in  the  complex  E.^  this  will  be  indicated  by*  E^  1  E.,. 

The  reducibility  used  by  B.  Peirce  is  defined  thus,  E  is  reducible^,  if 

E=^E,  +  E.  Ell  El  EI<Eo  EyE.,<Ey,  E.E^lE,. 

An  algebra  is  deleted  by  a  complex  E.^  if  the  units  in  E.^  are  erased  from 
all  expressions  of  the  algebra,  including  products.  The  result  is  a  delete 
algebra,  if  it  is  associative.     It  may  not  contain  a  modulus  however.^ 

134.  Theorem.    Let  the  product  of  ^a  be  given  by  the  equation 

l....r  r 

^0  =  2  a-i  yj  y^k  e^  —  2  x'^e^ 

ij,k  k  =  l 

If  the  units  may  be  so  transformed  that  the  product  may  be   expressed  by 
means  of  the  equations 

l....r' 

x'i  =  2  Xj  7/^.  y.j^.  1=1 r'  r'<^r 

i,k 
\....r 

x\,=  2  Xjy„Yi-J>'  i'=  r'  +  1 /• 

then   the  units  e^  .  .  .  .  c^,,  define  a  delete  algebra,®  called  hereafter  a  Molien 
algebra.     If  an  algebra  has  no  Molien  algebra,  it  is  quadrate. 

'See  refereuees  §  122.  'Scheffers  3,  4.  'Epsteen  and  Wedderbcrn  2. 

*EpsTEEN   and    WEDDERBrRN  2.     On    the  definitions   of   reducibility    see  Epsteex   and   Leonard  3; 

Leonard  2. 
'Scheffers  3,  4;    Hawkes  1,  3.     Cf.  Moi-ifn  1  ;    Shaw  5. 
•Molien  1.     This  is  Molieu's  "  begleitendes  "  system. 


46  SYNOPSIS  OP  LINEAR  ASSOCIATIVE  ALGEBRA 

135.  Theorem.  A  Molien  algebra  of  a  Molien  algebra  is  a  Molten  algebra 
of  the  original  algebra.  Two  Molten  algebras  which  are  such  that  the  co- 
ordinates of  the  numbers  of  the  two  algebras  have  q  linear  relations,  i.  e.,  whose 
numbers  are  subject  to  q  linear  relations,  possess  a  common  Molien  algebra 
of  order  q,  and  conversely.  If  the  Molien  algebras  of  an  algebra  have  no 
common  Molten  algebras,  then  the  numbers  in  the  different  Molien  algebras 
are  linearly  independent.' 

136.  Theorem.  If  the  complex  of  the  linearly  independent  numbers  of  the 
form  ^o  —  a^  be  deleted  from  an  algebra,  the  remaining  numbers  form  a 
commutative  algebra.^ 

137.  Theorem.  If  the  commutative  algebra  of  §  136  contains  but  one  unit 
the  original  algebra  is  a  quadrate.^ 

138.  Theorem.  If  the  delete  algebra  in  §  136  contain  more  than  one  unit 
it  may  be  further  deleted  until  the  delete  contains  but  one  unit.  This  unit 
will  belong  to  a  quadrate  algebra  which  is  a  delete  of  the  original  algebra.^ 

139.  Theorem.  The  scalar  of  any  number  contains  only  coordinates  which 
belong  to  the  units  in  the  commutative  delete  alg'  bra.^ 

140.  Theorem.  The  pre-  and  post-latent  functions  of  a  delete  algebra  are 
factors  of  the  corresponding  equations  of  the  original.  The  characteristic 
equation  of  the  delete  is  a  factor  of  the  characteristic  equation  of  the  original.^ 

141.  Theorem.  The  two  equations  of  a  quadrate  delete  algebra  are  powers 
of  the  same  irreducible  expression.' 

142.  Theorem.  An  algebra  is  a  quadrate  if  its  characteristic  equation  is 
irreducible  and  if  the  scalar  of  any  number  contains  only  coordinates  belonging 
to  the  units  of  the  quadrate  (which  may  be  a  delete  algebra).' 

143.  Theorem.  The  irreducible  factors  of  the  characteristic  equation  of  an 
algebra  are  the  characteristic  functions  of  its  delete  quadrate  algebras.' 

144.  Theorem.  The  number  of  units  of  a  delete  quadrate  is  the  square  of 
the  order  m,  of  its  characteristic  equation.     If  they  are  e^j,  then 

Cij e^i  =■  ^jk ea  i,  j,h,l=l m 

The  delete  quadrate  is  also  a  sub-algebra  of  the  original." 

145.  Theorem.    If,  in  a  SchefFers  algebra,  the  product  of  ^  into  and  by  the 

units  er,e,._, er_„,  vanishes,   provided  ^  is  not  a  modulus  or  a  partial 

modulus,  then  the  algebra  may  be  deleted  by  the  complex  of  e,. Cr  -  r,-    The 

'  Molien  1. 

'Molien  1.     Molien  points  out  that  tlie  unite  may  be  claasifled  according  to  their  quadrate  character, 
thus  approaching  Cartan's  theorem,  J  99. 


SUB-AIXJEBRAS.     REDUCIBILITY.     DELETION  47 

delete  algebra  will  have  an  equation  with  all  the  factors  of  the  original  algebra, 
but  each  appearing  with  an  exponent  lees  by  unity  for  each  deleted  direct  unit 
belonging  to  the  factor.' 

146.  Definition.  The  deficiency  of  a  Peirce  algebra  is  the  difference  between 
its  order  and  its  degree." 

147.  Theorem.  The  units  of  a  Peirce  algebra  may  be  so  chosen  that,  if  it 
is  of  deficiency  h,  one  unit  may  be  deleted,  giving  a  delete  algebra  of  deficiency 
h—  \,  which  is  a  sub-algebra  of  the  original.^ 

148.  Definitions.  An  algebra  E  is  semi-reducibJe  of  the  first  kind  when  it 
consists  of  two  complexes,  E^,  E.^^  such  that,'' 

E^Ey<E,  EiE,<E,  EoE^<Ez  E^E^iE, 

Am  algebra  is  semireducible  of  the  second  kind  when  it  satisfies  the 
equations* 

EyE^lE^  E\E,<E^  E^E^  =  Q  E^E^^E.^ 

If  in  any  algebra 

EiE^lE  EiE.,<E,^  EoEi<Es  E^E-^lE.;, 

then  Eo  is  called  an  invariant  suh-algebra} 

149.  Theorem.  \^  E  has  an  invariant  sub-algebra  E.,,  the  algebra  K  pro- 
duced by  deleting  E.,  is  a  delete  of  E,  called  con^plementar y  to^  E^. 

150.  Theorem.  If  ^,  is  a  maximal  invariant  sub-algebra  oi  E,  and  if  there 
exists  a  second  invariant  sub-algebra  E.,  o{  E,  then  either  E  is  reducible  or  E.^ 
is  a  sub-algebra  ^  of  Ey. 

151.  Theorem.  If  £",  and  E.,  are  maximal  invariant  sub-algebras  of  £",  and 
if  E^o  -^  0,  then  Ey^  is  a  maximal  invariant  sub-algebra  of  both  Ey  and  E.^.^ 

152.  Theorem.  A  normal  series  of  sub-algebras  of  E,  is  a  series  Ei,  Eo, .... 
such  that  jB",  is  a  maximal  invariant  sub-algebra  of  £"8  _  J  (-ffo^-E").  lfKi,Ko,.  .  .  . 
are  the  corresponding  complementary  deletes,  then  apart  from  the  order  the 
series  /fj,  K,,  ....  is  independent  of  the  choice  of  J^i,  E^,  . . .  .* 

153.  Theorem.  Let  «,  be  the  order  of  ^,;  7,,  the  difference  between  aj_i 
and  the  maximal  order  of  a  sub-algebra  of  £"4 _j  which  contains  iS", ;  /.•,  =a,_j  —  a,. 
Then  the  numbers  /j,  L,  ....  are  independent  of  the  choice  of  the  normal  series 
apart  from  their  order.     A  like  theorem  holds  for  l-i,  K,  .  .  .  .* 

154.  Definition.    An  algebra  which  has  no  invariant  sub-algebra  is  simple} 


>S0BEFFER3  3.  >  StaEKWEATHBR  1.  >  EPSTBKS  1,  3. 

*  Epsteen  1.  s  Epsteen  and  Weddbrbcrn  2. 


48  SYNOPSIS  OF  LINEAR  ASSOCIATIVE  ALGEBRA 

155.  Theorem.    The  complementary  deletes  K^,  K^,  .  .  ■  ■  are  all  simple.' 

156.  Definition.    The  series  E,  Pj,  Po,  ....  is  a  chief  or  principal   series 
when  Pj  is  a  maximal  sub-algebra  of  Pg_i  which  is  invariant'  in  E. 

157.  Theorem.    The  system  of  indices  of  composition  is  independent  of  the 
choice  of  the  chief  series,  apart  from  the  sequence.^ 

158.  Theorem.    An  algebra  is  irreducible  if  its  quadrates  may  be  so  arranged 
Qi,Qz-  •  •  •  Qp  that  there  are  skew  units  of  characters  (21),  (32)  ... .  (^^jp  —  1).^ 


VI.  DEDEKIND  AND  FROBENIUS  ALGEBRAS. 

159.  Definition.  A  Dedekind  algebra  is  one  which  is  the  sum  of  quadrates 
Qi,  Qz Qi,-     Its  order 3  is  r  =  ?q  + +  "7, • 

160.  Theorem.  A  Dedekind  algebra  has  a  sub-algebra  of  order  A,  whose 
numbers  are  commutative  with  all  numbers  of  the  Dedekind  algebra.  No 
other  numbers  than  those  of  this  sub-algebra  are  so  commutative.* 

161.  Theorem,  A  Dedekind  algebra  is  reducible  and  the  sub-algebras  are 
found  by  multiplying  by  the  numbers  e„,  a  =  1  .  •  ■  •  /;,  in  terms  of  which  the 
commutative  sub-algebra  may  be  defined,     [e^e^  =  S^^^e^].^ 

162.  Theorem.  The  characteristic  equation  of  a  Dedekind  algebra  is 
Ai  A.  .    .  .  A„  =  0.     The  pre-  and  post-equations  ^  are  AJ''  A^"= AJf"  =  0. 

163.  Theorem.  If  a  Dedekind  algebra  has  only  linear  factors  in  its  equation 
it  is  a  commutative  algebra.^ 

164.  Theorem.    The  scalar  of  e„  is  given  by  the  equation 

The  scalar  within  a  single  quadrate,  Qi,  may  be  indicated  by  S.^.     For  any 
number  we  have  ^ 

i 

165.  Theorem.  An  algebra  is  a  Dedekind  algebra  when  in  the  general 
equation,  vu,  the  coefficient  of  ^''  ^,  contains  each  coordinate  in  such  a  way 
that  the  equations 

0  1=1 


dm^^ 


give "  a:i  =....=  ar,  =  0 


>  Epsteen  and  Weddekdukn  3.  sScueffers  8,  4. 

»Cf.  Frobesius  14.     Cabtan  2.     This  la  Cartau's  semi-simple  algebra. 

<  FKonENirs  14.     He  cails  tliese  invariant  numbers.  ''  Frobenius  14. 

•Cautan  3,  see  §121.     Evidently  |  m,  (e.,  «^)  |  ±0. 


DEDEKIND  AND  FROBENIUS  ALGEBRAS  49 

166.  Theorem.  If  Aj  is  the  deluniiiuant  shear  factor  corresponding  to  the 
quadrate  Q,,  then  Si  .  A,  =  0  for  all  numbers  of  the  algebra,  and  if  e,  is  the 
partial  modulus  of  this  ([uadrate,* 

ei  Ai  =  A.  <-!  =  0 

The  i  +  1  scalar  coefficient  of  any  numbers  vanishes  ;  i.  e. 

167.  Theorem.    If  A,  («)  =  Aj  [b)  then  for  a  determinate  number^  c 

c~^ac  =  /j 

168.  Definition.  A  Frobenius  algebra  i.s  one  which  can  be  defined  by  r 
numbers  o^  .  ■  .  ■  o,  which  .satisfy  the  equations 

oT'  =  ^0  =  Oi  i=l r 

Oi  Oj  =  Ok  ur^o^  =  Oj  Oi  =  o^oj-  •  hj=^ r 

o, Oj  .  0,,  =  0i  .o, o„  i,  j,k=\ r 

The  multiplication  table  of  these  units  defines  a  group,  and  any  group  of  finite 
order  or  infinite  order  may  be  made  isomorpl)ic  to  a  P'robenios  algebra.'' 

169.  Definition.    Two  units  o,,  Oj  are  amjugate  if  for  some  determinable 

unit  0^. , 

Oi  =  o„Ojor^ 

If  we  operate  on  Oj  by  all  units  of  the  algebra,  Oj  .  .  .  .  o^,  we  arrive  at  r 
different  units  as  results.  These  are  said  to  constitute  the  ^th  conjugate  class. 
There  will  be  k  of  these  classes.     Also  /^  is  a  divisor  of  r. 

170.  Theorem.  For  each  unit  in  a  conjugate  class  we  have  (as  Oj  is  the 
modulus  or  not) : 

S  .  0^  Oj oi~^  =  S  .  OjZ=  1  or  0 

171.  Theorem.  If  the  sum  of  all  the  units  in  the  ^th  conjugate  class  be  Ix',, 
then  for  any  unit 

A'(  Oi  =  o,Kt  i=l r 

There  are  k  different  numbers  Kf,  K^  . . .  .  K,,. 

172.  Theorem.  The  Ic  numbers  K^,  <  =  1  . ...  1c  constitute  a  commutative 
algebra  of  A;  dimensions,  that  is 

173.  Theorem.    We  have  (according  as  o<  is  not  or  is  the  modulus) : 

S  .  Kt  =  -^  S  .  lOi  Of  of  ^  =  rtS  .o,  =  Q  or  r^ 

174.  Theorem.  A  Frobenius  algebra  is  a  Dedekind  algebra  of  k  quadrates. 
The  k  numbers  Ki  determine  the  k  partial  moduli,  one  for  each  quadrate. 


'  FaoBENins  14.     Shaw  4.  =  Frobenius  i4.     Other  theorems  appear  in  Chapter  XIX,  Part  II. 

sFrobenicsS,  4,  5,  6,  7,  8,  9,  10,  11,  12,  18;   Dickson  I,  2,  3,  4;  Burnside  1,  5;  Poincabe  4 ;  Shaw  6. 


50 


SYNOPSIS  OF  LINEAR  ASSOCIATIVE  ALGEBRA 


The  widths  of  the  quadrates  being  represented  by  w,,  i  =  1  ....  k,  we  have 

Kt^'^gu^i  i,t=l k 

175.  Theorem.    It  follows  that  if  we  take  scalars 

r  .  SKi  =  r  =  wl  +  ii4 +wl 

r  .  SKj  =  0  =  wlgji  +  tdgj3 +  m9jk        i  =  2 k 

176.  Theorem.    Let  the  scalar  of  o<  in  the  quadrate  i  be  represented  by 
AS'foj,  orsf,  then 


k 

2 

i  =  l 


^.0,=  2  M;f./S<>,=  2  toW? 


k 

2 


177.    Theorem.    We  have 

^«  K,  =  g,,  ^«  e,  =  g,,  =  r,  .  S'^  o,  =  r,  *'« 
Hence  ^  _  ^^  ^(^  +  m,|  ««>+....  +  li'X «f ' 

0  =  «-js;.'>  +  M;|«f  +....+  <4«r  j=2....k 

If  we  write  for  icj  sj"'  the  symbol  xf}   (called   by   Frobenius   the  i-th 
characteristic  of  Oj)  we  have 

_  -.  _A  _   fi) 
"'J  —  ^  ^  —  Xi 

where  A  is  the  determinant  \x\\  xf  ■  ■  ■  ■  X<*'l  ^"*^  ^j  '^  ^^^  minor  (including 
sign)  of  tc^.     This  determinant  A  evidently  cannot  vanish. 


Also 


Ki  =  y-t  2  e^ 


<=i 


Xt 


rt'Zeis'i 


(t) 


anc 


(2^,f=r2ir, 
178.    Theorem.    Hence 


s?   K, 

/rj  .  .  .  .  «[*■' 

.  .  .  si*' 

4"    K„lr^....sf^ 

-=- 

41'    S«>       . 

•  •  •  sf 

4"  ^./n-  •  •  •  ■  «lf^ 

4'>  4^'     • 

...sr 

MJ, 

;ti'^  ;tf  •  •  ■ 

or 
or 


179.    Theorem.    For  all  values  of  a,  h 

d  =  l  ^h 

rs'^  6'«'  =  2  S^'^ 0,0^0,0;^' 

d  =  l 


—  -y*"'  y«>  =  to,  2'  y"> 
~  Xa  Xl>  —  "'i  -^  Xac 
'  b  c 

where  2'  takes  o^  over  the  rj,  values  in  the  conjugate  class*  of  0^. 

c 

Also' w,wj  8^^  s^J^=  2  rf,^C'  or  x"^  x'i'  =  2  ci'i>;ta' 


u=l 


«  =  1 


'  See  references  to  §  168.     These  apply  to  theorems  following. 


'BCRNSIDE  5. 


DBDEKIND  ANIJ  FROBENIUS  ALGBBKAS  51 

180.  Theorem.        2  .  A""  .  o^o^^  S"'  o^  =  -^  ^'^  o, 

b=i  ^i 

181.  Theorem.        k  .  A"" .  o^o^^  A"^' o,,  =  0  t  :^:y 

b=l 

182.  Theorem.        S  /S'<«  o^  .  S'^^  0,7^  =  -4- 

183.  Theorem.        i  *9">  o„  o;;'  *S""  o,  o,,  =  - ,-  /S"«  .  o„  o, 

h=i  "'i 

184.  Theorem.        2  /S'">.  o"'  oz^  o„  o^  =  -3- 

a,fi=l  ^i 

185.  Theorem.  If  o,  i.s  an  independent  generator  of  the  group  of  units, 
Off  ...  .  Or-i)  '•'^^^  ^^  we  form  the  t-th  LaGrangian  of  Oj,  that  is, 

where  u  is  a  primitive  mj-th  root  of  unity,  and  »ij  is  the  order  of  Oj  {o^'  =  o^) 
then  for  any  number  of  the  algebra,  ^,  we  have  a  product 

such  that  all  numbers  of  the  algebra  are  separable  into  wJ]  mutually  exclusive 
classes  of  the  forms  (where  it  is  sufficient  for  ^  to  be  any  one  of  the  units  Oj 
when  the  group  is  written  in  the  form  OjO\). 

Uu  {t  =  \  ....m,) 

For  ^/it,  we  have* 

186.  Theorem.  If  0.,  is  a  .second  independent  generator,  then  we  may 
determine  the  equations  oi  o.;^  f^i  {t  =■  1  .  .  .  .  m^.  The  latents  Zi,  determined 
as  in  §48,  used  as  right  multipliers,  separate  the  numbers  of  the  algebra  into 
mutually  exclusive  classes,  such  that  if  these  latents  are  /^^^^ ,  tlien  (if 
u  :{:  u',  t  4:  t') 

Sfuufuu   -—  ^/uu  S/Ku  fu'u'  ^—  0 

This  process  of  determining  latents  by  the  independent  generators  may  be 
continued  until  they  are  in  turn  exhausted. 

187.  Theorem.  The  ultimate  latents  are  scalar  multiples  of  independent 
iderapotents  of  the  forms  /IJ'J,,  where  i=  1  .  .  .  .u'l;  s  ^=  1 .  .  .  .A:  Multiplication 
right  and  left  by  these  idempotents  will  determine  every  quadrate  unit  P.,^^, 
i,  y  :=  1  .  ■  .  ■  u\;  s  =  1  ....  A",  in  terms  of  the  c  generators  Oj  .  .  .  .  o^. 

188.  These  results  may  be  extended  easily  to  cases  in  which  the 
coefficients  of  the  units  Oj  are  restricted  to  certain  fields. 

I  Shaw  6.     This  reference  applies  to  §S  186,  187. 


52  SYNOPSIS  OF  LINEAR  ASSOCIATIVE  ALGEBRA 

Vn.     SCHEFFERS  AND  PEIRCE  ALGEBRAS. 

189.  Theorem.  Every  SchefFers  algebra  with  h  partial  moduli  has  h  sub- 
algebras,  each  with  like  pre-  and  post-  characters. 

190.  Theorem.  The  general  equation  of  a  Scheffers  algebra  of  h  partial 
moduli  is  of  the  form' 

n  .  (aj  —  <f)"ii  +  "i2  +  ••••  +  «ft  =  0 

191.  Theorem.  Every  number  of  a  Scheffers  algebra  satisfies  the  general 
equation  of  its  direct  sub-algebra,  which  is 

n  (a,-0"«  =  o 

This  equation  is  the  intermediate  equation  of  the  algebra. 

192.  Theorem.    The  characteristic  equation  of  a  Scheffers  algebra  is 

n  (a,-0'''  =  0 
1=1 

193.  Theorem.    A  Peirce  algebra  may  have  its  units  taken  in  the  form^ 

cj<^  s  =  \  ....p      t  =  0....  ^,-1 

194.  Theorem.  Units  containing  6',  t  >  0,  may  be  deleted,  and  the 
remaining  numbers  will  then  form  a  companion  delete  algebra,  called  the  base 
of  the  Peirce  algebra.^ 

195.  Theorem.  Any  Peirce  algebra  may  be  made  to  serve  as  a  base  by 
expressing  its  units  in  terms  of  associative  units  of  weight  zero.^ 

196.  Theorem.    The  product  of  two  units  follows  the  law  ^ 

C    ff'    C      fit''  —  y/i  C      fit'"  /"'  >  fl  A-  i'l 

197.  Theorem.  A  Peirce  algebra  of  order  r,  degree  r,  is  composed  of  the 
units '' 

These  have  been  called  by  Scheffers,  Study  algebras. 

198.  Theorem.  A  Peirce  algebra  of  order  r,  degree  r —  1,  is  composed  of 
the  units 

Cj  =  Xlio  +  >^220       ^S  — -  '^aiO  "H  "^12r-2       ^3  ^  '^111  H"  "'^IS  r— 2       ^4  — "  ^^112  •  •  •  •  ^r -— '^ll  r-2 

'Cartan2.         »Siiaw5.     Of.  Strono  I.         ^SiiawS.         •'Shaw  5.    Cf.  Sciikffers  3 ;  Caktan  3. 
'  B.  Peikce  3  ;    ScheffersS;    Hawkks  1 ;    Shaw  5;    Stddy  3. 


SCHEFFERS  AND  PEIRCB  ALGEBRAS 


53 


This  is  reducible,  if  a  and  b  do  not  vanish,  to  the  case  of  a  =  1  =  /--. 
If  a  =  0,  we  may  take  b—\  or  0.     If  h  —  0,  we  may  take  a  =  1  or  0. 
When  r  =  4,  eitlier  a  =  1,  />  has  any   value  ;  or  a  =  0,  /^  =  1  ;  or  a  =  0, 
?>  =  0. 

If  r  =  3,  a  =  0,  h  =  0.' 


199.    Theorem.    A  Peirce  algebra  of  order  r,  degree  r —  2,  is  of  one  of  the 
following 
case 


The  ?.  wi 

When 

(1).  ^0  = 
(11).  e,= 

(12).  ej  = 
(13).  e,= 

(14).  e,= 

(2).  e„  = 
(21).  e,= 

(22).  ei  = 

(23).  ej  = 
63  = 

(24).  e,= 

63  = 

(25).  e,  = 

(26).  e,  = 

(27).  e,= 

(?8).  e,= 


(29).  e 


ypes."     Only  the  forms  of  e^,  e,,  e.,  eg,  e^  are  given  since  in  every 

^5  =  '^ns   •  •  •  •    ^r-2  ==  ''•11  r-4  ^r  - 1  =  ''-H  r  -  3 

be  omitted  iu  each  case. 

r>  6. 

110)  +  (220)  +  (330),  type  of  algebra  (*,  ^^y,/ /-^) 

210) +  (320)  + (13  r  — 2)        e,  =  (310)  +  (12  r  —  2)        e3  =  (lll) 

e,  =  (112) 
210) +  (320)        ('3  =  (310)        e3=(lll)  ^^  =  (112) 

210)  +  (320)  + (13  r  — 2)        c„  =  (310)+    (12  r— 2) 
111)+  2(13  r  — 2)  e,  =  (112) 

210)  +  (320)       e,=  (310)        63  =  (111)  +  2  (13  r- 2)     ei  =  (112) 

110)  +  (220),  type  of  algebra  {i,j,  ij,f j''^) 

210)  +  (12  r— 3)       e.  =  (211) +  (12  r— 2)       eg  =  (ill)  +  (221) 

e,=  (112) 
210)  +  (12  r— 3)  6,  =  (211)  + (12  r  — 2) 

111)  +  (221)+  2(12r—  2)  ^^  =  (112) 

210)  +  (12  r  — 3)  e2  =  (211)  +  (12  r— 2) 

lll)  +  (22l)  +  2(12r— 3)+2c(12r— 2) 

64  =  (112)  + 4  (12  ?•— 2)  c  =  0ifr4:8 

210)  +  (12  r— 3)  +  (12r— 2)       Co  =  (211)  —  (12  r  — 2) 
HI)  — (221)  — 2(12>-— 3)  e4  =  (112) 

210)  + (12 '-  —  3)       e,  =  (211)  — (12r— 2)       63  =  (HI)  —  (221) 

.       e,  =  (112) 
210) +7i(12r— 2)     63=  (211)  h=0  or  l'\i  r:^l 

lll)  +  2ic(221)  +  2  (2-c)  (12r— 3)       e,  =(112)  +  4  (11  r— 2) 
210)  +  (12  r— 3)        6,  =  (211)  — (12  r— 3)        63  =  (111)  — (221) 

e,  =  (112) 
210)  + A  (12;-— 2)     e,=  (211)      63  =  (ill)  +  (221)  +  2  (12  r  — 3) 

64  =  (112)  A=0orlifr:^7 

210)  +  (12  r— 2)       Co  =  (211)      63  =  (111)  +  rf  (221)      e,  =  (112) 


'B.  Peirce  3;    Scheffers  3  ;    Shaw  5. 

4 


2  Starkweather  1.     Cf.  Shaw  5. 


54 


SYNOPSIS  OF  LINEAR  ASSOCIATIVE  ALGEBRA 


(2a).  ei  = 
(2y).  e,= 
(2f).  e,  = 

(27,).  e,  = 

(3).  Co  = 
(31).  e,= 
(32).  ei  = 
(33). 
(34). 


e,  = 


ci  — 


e,  = 


(35) 
(36).  e,= 


(37).  .,= 
(38).  e,  = 
(39).  e,  = 


210)      6,=  (211)      e3  =  (lll)  +  c?  (221)  C4  =  (112) 

210) +  (221) +  (12  ;•— 2)   6.  =  (211)    e3=(lll)  64  =  (112) 

210) +  (221)             62=  (211)   e3  =  (lll)  e<=(112) 

210)      e,  =  (211)      e3=(lll)  +  2(12r— 2)  64=  (112) 

2l0)  +  (12r— 2)  e2  =  (211)  fg  =  (lll)  +  2  (12?— 2)  e4=(112) 

210) +  (12  ?•— 2)  e2  =  (21l)  e3  =  (lll)  ei=(112) 

210)            6,=  (211)  e3  =  (lll)  e4  =  (112) 

110)  +  (220)  +  (330),  type  of  algebra,  {i,j,  k,  1c . . . .  kT'^) 

2l0)  +  (12r— 2)               63=  (310)              63  =  (111)  e4  =  (112) 

210)                                    62  =  (310)              63  =  (111)  f4  =  (112) 

210)  +  g{lSr-2)            62  =  (310)  +  (1  2  r— 2)    e3=(lll)  e,  =  (ll2) 
2l0)  +  (13r— 2)               e2=(310)  +  (127--2) 

111)  +  2(12r— 2)  +  2(13r— 2)  e4=(ll2) 
210)  +  (13r— 2)               e2=(310)  +  (l2r— 2) 

<'3=(lll)  +  2(l3r  — 2)  e4  =  (112) 

210)  +  (12?— 2)— (13r— 2)        63  =  (310)  +  (12  r— 2)  e3  =  (lll) 

e,  =  (112) 

e3  =  (lll)+2(l3r— 2)  e4  =  (112) 
63=  (111)  + 2  (12  r— 2) 
63=  (11 1)  + 2  (13  r— 2) 


e,  =  (112) 
e,  =  (112) 


210)  +  (l2r— 2)  6,  =  (310) 
210)  +  (12r— 2)  Co  =  (310) 
210)  e2  =  (310) 

When  r  =  4,  5,  or  6.     These  cases  may  be  found  in  XX. 

200.  Theorem.    A  Scheffers  algebra  of  degree  r —  1,  which  is  not  reducible, 
must  consist  of  two  Study  algebras,  with  one  skew  unit  connecting  them.^ 

201.  Theorem.    A  Scheffers  algebra  of  degree  r  —  2,  which  is  not  reducible, 
must  consist  of 

(A)  Three  Study  algebras,  E^,  E.,  E^,  with  skew  units  (12),  (23); 

(B)  One  Study  algebra,  and  one  algebra  of  deficiency  unity,  with  one 

skew  unit  connecting  them  ; 

(C)  Two  Study  algebras,  joined  (1)  by  two  skew  units  (12)  (12),  or 

(2)  joined  by  skew  units  (12),  (21). 

Theorem.    A  Peirce  algebra  whose  degree  is  two,  is  determined  as 
2r— 1 


202 
follows : 


for  m 


V8 


we  may  take 


Cj,  Cj  . . . .  e,„ ^ £'1,  such  that  EE^  :=  E^E  =  0 
The  remaining  units  are  such  that 

«m  + 1  em  +  ^  =  ^Ym  +  i.m+j, k  Sk  ^ 0  =  modulus,  k=l  .. 

or  in  brief 

E=E,^E„  E\  =  0,  E,E,  =  E,E,  =  0,  E\<E„  ^i^j=-^j^i 


m 


>  SOHErrsiis  8. 


SCHEFFERS  AND  PEIRCE  ALGEBRAS  65 

One  class  of  Peirce  algebras  of  degree  two,  and  order  r,  may  be  con- 
structed from  the  algebras  of  degree  two  and  order  less  than  r,  by  adjoining 
to  the  expressions  for  the  algebra  chosen  for  the  base  other  terms  as  follows: 
let  the  units  of  the  base  be  e«  .  .  .  .  e^.  .  .  . .  written  with  weight  zero,  say  Cjo,  e^] 
then  the  adjoined  unit  (deleted  unit)  being  e^-i  =  ?t,ii,  we  have  for  new  units 

^10  ^^  ^iO  +  ^«  '^121  +  ''(3  '''131  +    •  •  •  • 
^jO^^^JO"^  ^J2  ^m  M"  ^13  ''-131  +    •  •  •  • 

and  Uij  =  —  a_,(  for  all  values  of  i,  j. 

The  second  and  only  other  class  involve  units  of  forms  ;i,ii  +  .  .  . .  and  are 
given  by 

e^  =  ^io  +  "gi'  '^121  + +  «2a  ''-221  + 

CjQ  =  «^0  +  O^  ^121  +       •  +  «^'  ^221  + 

^.  =  '^m  —  ^1  '^221  —  ^2  ''-331  •  •  •  •  ^1,  §2  •  •  ■  •  =  0  or  1 

and  afi^  =  —  a^'  for  all  values  ^  of  i,  j\  h. 

203.  Theorem.    A  Scheffers  algebra  of  order  r,  degree  two,  consists  of  two 

partial    moduli    ;i|,o  +  ^220  +   •  •  •  •   +  '^m,n.,o  and  ^m,  +  i.m,  +  i.o  + +  ''■r^, 

and  r —  2  skew  units  as  follows" 

'^Wi  +  2  10. . . .  '^rlO  '^2  Wi  +  1  0  '^3  nil  +  1  0  ... .  '^-mi  m,  +  1 0 

204.  The  subject  of  the  invariant  equations  of  Peirce  and  Scheflfers 
algebras  is  under  consideration.     Some  particular  cases  are  given  later. 

'Shaw  5.  ^Scheffebs  3. 


56  SYNOPSIS  OF  LINEAR  ASSOCIATIVE  ALGEBRA 

Vm.     KRONECKER  AND  WEIERSTRASS  ALGEBRAS. 

1.     KRONECKER  ALGEBRAS. 

205.  Definition.  A  commutative  algebra  is  one  such  that  every  pair  of 
numbers  i^,,  ^j  in  it,  satisfy  the  equation  :^ 

206.  Theorem.    An  algebra  is  commutative  when  its  units  are  commutative. 

207.  Theorem.  The  characteristic  equation  of  a  commutative  algebra  can 
contain  only  linear  factors,  if  the  coordinates  belong  to  the  general  scalar  range. 

208.  Theorem.  If  the  characteristic  equation  of  a  commutative  algebra 
whose  coordinates  are  unrestricted  has  no  multiple  roots  it  is  reducible  to  the 
sura  of  r  algebras  each  of  one  unit,  its  partial  modulus.  Such  algebra  is  a 
Weierstrass  algebra.' 

209.  Theorem.  If  the  characteristic  equation  of  a  commutative  algebra 
has  p  distinct  multiple  roots,  it  is  reducible  to  the  sum  of  p  commutative 
Peirce  algebras.     Such  algebra  is  a  Kronecker  algebra.^ 

210.  Theorem.  The  basis  of  a  commutative  Peirce  algebra  is  a  commuta- 
tive algebra. 

211.  Theorem.  A  Kronecker  algebra  may  contain  nilpotents,  a  Weiek- 
STRASS  algebra  can  not  contain  nilpotents.*  A  Weierstrass  algebra  has 
nilfactoriaLs. 

212.  Theorem.  If  the  coefficients  are  restricted  to  a  range,  such  as  a  field 
or  a  domain  of  rationality,  the  algebra  may  not  contain  either  nilfactorials  or 
nilpotents.  Such  cases  occur  in  the  algebras  built  from  Abelian  groups.  This 
case  leads  to  the  general  theorem :  If  the  equation  of  the  algebra  is  reducible 
in  the  given  coordinate  range,  into  p  irreducible  factors,  the  algebra  is 
reducible  to  the  sum  of  ^j  algebras  and  there  are  nilfactors.  Each  irreducible 
factor  belongs  to  one  sub-algebra.  If  an  algebra  has  an  irreducible  equation 
in  ^,  the  general  number,  such  that  the  resolvent  of  this  equation  and  its  first 
derivative  as  to  ^  does  not  vanish,  then  all  its  numbers  ma}-   be  brought  to 

the  form 

'(=b,  e,  +  b,  i  +  b,i~  +  bsP+  ....  +  K_,  i'-' 

where  i  is  a  certain  unit  of  the  algebra,  and  i„  .  .  .  .  i^-i  belong  to  the  range. 
If  the  resolvent  vanishes  for  either  a  reducible  or  an  irreducible  equation, 
there  are  uilpotent  numbers  in  the  algebra.^ 


'References  for  certain  commutative  algebras  follow  in  the  next  article.     On  the  general  problem  see 
Stody  2;    Fkobbnics2;    Kroneckeu  1  ;    Shaw  4. 
»Sce  references  for  §21.5,  also  KuoNF.cKEn  1. 

>  MOOBE  1.  *  KllONECKER  1.  '  MOOBE  1  ;     KnONBCKER  1. 


KRONECKER  AND  WEIERSTRASS  ALGEBRAS  57 

213.  Theorem.    In  canonical  fornn  the  adjoined  unil  is  of  form 

j=  k  K.i  +  1  K.,+  ■■■■ 

«  =  1  »  =  2 

Tiiere  are  as  many  terms  of  a  given  weight  k  as  there  are  Vjasal  units 
with  subscripts  that  appear  in  terms  of  weight  k. 

214.  Theorem.  The  units  of  a  commutative  Peirce  algebra  may  be  taken 
of  the  form. 

^1    S2    •  •  •  •  Cm 

where  t,  =  0  ....  jIa,;  and  where  ^j"^',  for  i<;  m,  is  linearly  expressible  in 
terms  of  higher  order. 

2.     WEIERSTRASS    ALGEBRAS. 

215.  Definition.  A  Weierstrass  algebra  is  a  commutative  algebra  satisfying 
the  conditions  ^'i^j=^jCi  ^"^  whose  degree  equals  its  order,'  and  whose 
coordinates  are  real. 

216.  Theorem.  Numbers  whose  coefficient  wi^  =  0  are  nilfactorial  ("divisor 
of  zero").  The  product  of  a  nilfactor  and  any  number  is  a  nilfactor.  There 
are  no  nilpotents  in  the  algebra." 

217.  Theorem.  Tliere  is  at  least  one  number  (j,  such  that  e^,  g,  g- .  ■  ■  ■g'~^ 
are  linearly  independent.  The  latent  equation  resulting  may  be  factored 
into  r  linear  factors,  the  imaginary  factors  occurring  in  conjugate  pairs. 

218.  Theorem.  A  Weierstrass  algebra  is  reducible  to  the  sum  of  r'  algebras 
of  the  form 

Xi         A=i<-i         ^iXj  =  ^         i,j=l.--    r'         r  =  r'-\-r"         i^J 

and  whose  coordinates  are  scalars,  which  appear  in  conjugate  forms  if 
imaginary  (/■"  is  the  number  of  algebras  admitting  imaginaries).  Hence  the 
algebras  may  be  taken  to  be  of  the  form 


Xi+Xi  +  l  (Xi—  X;  +  i)V—   1 

with  real  coefficients ;  or  finally  we  may  take  the  r'  algebras  as  r'  independent 
ordinary  complex  algebras. 

219.  Theorem.  Nilfiictors  are  numbers  belonging  to  part  only  of  the 
partial  algebras.  If  ^1,2. ...,i  has  coordinates  in  the  first  n  algebras  but  not  in 
the  other  r' — ?i,  ^„  +  i....r  ^^  coordinates  only  in  the  algebras  from  the 
?i  +  1-th  to  the  ?-'-th,  then 

Sl  ....  n  Sn  +  l....r'  ^^  0 

'  Weierstrass  2 ;    Scbwarz  1;    DedekindI,  2;    Bbrlott  1 ;    Holder  1;    Peterson  2;    Hilbbrt  1; 
Stolz  1 ;   Chapman  3.     The  sections  below  are  referred  to  Berloty. 
'The  preaeuce  of  nilpotents  would  loner  the  degree. 


58  SYNOPSIS  OF  L.INEAK  ASSOCIATIVE  ALGEBRA 

IX.     ALGEBRAS  WITH  COEFFICIENTS   IN   ARBITRARY   FIELDS. 
REAL  ALGEBRAS.     DICKSON  ALGEBRAS. 

220.  Definition.  An  algebra  is  said  to  belong  to  a  certain  field  or  domain 
of  rationality,  when  its  coordinates  are  restricted  to  that  field  or  domain.  In 
particular  an  algebra  is  real,  when  its  coordinates  are  real  numbers.^  The 
term  "finite"  algebra  is  used  also  to  mean  algebras  whose  coordinates  are  in 
an  abstract  (Gralois)  field. 

221.  Theorem.  The  coefficients  of  the  characteristic  and  the  latent  equa- 
tions of  an  algebra  are  rational  functions  of  the  coordinates  in  the  domain 
^{x,y)}  which  is  the  domain  of  the  coordinates  and  the  constants"  y. 

222.  Theorem.  If  new  units  are  introduced  by  a  transformation  T' rational 
in  £l^,  the  new  units  are  rational  in  £1^,;  the  hypercomplex  domain  ^ix,e)  is 
then  identical  with  the  hypercomplex  domain  n,^r_e')-  Further,  if  £1^  contains 
£ly,  it  also  contains  £L^,. 

223.  Theorem.  If  /S'.  ^  is  defined  for  any  domain,  then  S  .  1^,1%  invariant 
under  any  transformation  of  the  units  of  the  algebra  and  is  rational  in  Hx.y 

224.  Theorem.  In  any  domain  there  is  an  idempotent  number  or  all 
numbers  are  nilpotent. 

225.  Theorem.  In  a  Peirce  algebra  every  number  ^  =  ^o  +  ^i,  where  ^^  is 
a  multiple  of  the  modulus,  and  ^j  is  a  nilpoteut  rational  in  Ilj.^.  This 
separation  is  possible  in  only  one  way.  We  may  choose  by  a  rational  trans- 
formation new  units  such  that 

e«  =  e^  eT'  =  0  i=\  ....r-\ 

The  characteristic  equation  of  ^  is  F .  C,=-t,''  [-^i^]'',  where  F ,  ^  is  rationally 
irreducible  in  £1^^. 

226.  Theorem.  In  any  Scheffers  algebra,  we  may  choose  by  transforma- 
tions rational  in  H^^,  the  units  y;  which  are  nilpotent  such  that 

227.  Definition.  A  real  algebra  may  be  in  one  of  two  classes,  the  real 
algebras  of  the  first  class  are  such  that  their  characteristic  equations  have  no 
imaginary  roots  for  any  value  of  ^,  the  general  number;  the  second  class  are 
such  that  their  characteristic  equations  in  ^  have  pairs  of  conjugate  imaginaries.' 

228.  Theorem.    Every  real  quadrate  is,  if  in  the  fii'st  class,  of  the  form 

6'^  ^"  '^  (1 )     <-,;■  e,i  =  ^jke,i  iz=i....  J) 

If  of  the  second  class,  it  is  of  order  4j/,  and  is  the  product  of  Q  and  an 
algebra  of  the  first  class  (l). 

'Dickson  5;    Tabek  4.     Hamilton  restricted  Quaternions  to  real  quaternions,  calling  quaternions  with 
complex  coordinates,  biquaternions. 

'Taber  4.     The  succeeding  sections  are  referred  to  Taber  4.     This  paper  contains  otber  tbeorems. 
3CARTAN  2.     This  reference  applies  to  §§228-232. 


ALGEBRAS  WITH  COEFFICIENTS  IN  ARBITRARY  FIELDS  59 

The  algebra  Q  is  Quaternions  in  the  Hamiltonian  form 
e^,  h  3,  k,  ij  =  —ji  =  /c,  etc. 

229.  Theorem.  Every  real  Dedekind  algebra  is  the  sum  of  algebras, 
each  of  which  is  of  one  of  the  following  three  types  : 

(1)  Real  quadrates  of  first  class; 

(2)  Real  quadrates  of  second  class  ; 

(3)  The  product  of  a  quadrate  of  first  class  and  the  algebra  eg,  e,,  where 

230.  Theorem.  Every  real  Scheffers  algebra  of  the  second  class  is  derivable 
from  one  of  the  first  class  by  considering  that  each  partial  modulus  belonging 
to  a  complex  root  of  the  characteristic  equation  will  furnish  two  units  for  the 
derived  algebra,  say 

gj  =:  Xj  +  Xj  ^2  =  ('f  1  —  ^2)  *^  —  1 

That  is,  the  direct  sub-algebra  consists  of  direct  nil-potent  units  and  of  the 
sum  of  algebras  of  the  forms 

Co   or   eo,ej  (^1  =  — Cq) 

All  other  units  are  chosen  to  correspond ;  thus  r^^^  furnishes  two  units,  rj^  and 
>7.^,  corresponding  to  x^i  *^ —  1  ^i' 

231.  Theorem.  A  Cartan  real  algebra  is  primary,  and  has  a  Dedekind 
sub-algebra  according  to  §229,  the  other  units  conforming  to  this  sub-algebra 
in  character,  and  giving  multiplication  constants  y  which  are  real ;  or  it  is 
secondary,  and  has  a  Dedekind  sub-algebra  consisting  of  the  algebras  in  §229 
multiplied  by  real  quadrates  of  the  first  class,  the  other  units  conforming  as 
usual. 

232.  Theorem.  Every  real  irreducible  (in  realm  of  real  numbers)  com- 
mutative algebra  is  of  the  types  of  §230.  It  is  a  Peirce  algebra  then,  the 
modulus  being  irreducible;  or  else  it  has  two  partial  moduli  which  give  an 
elementary  Weierstrass  algebra,  and  hence  are  irreducible  in  the  domain  of 
real  numbers. 

233.  Theorem.  The  only  real  algebras  in  which  division  is  unambiguous 
division  (in  domain  of  reals)  are  (1)  real  numbers;  (2)  the  algebra  of  complex 
numbers    eg  =  ^o  =  —  ^      ^j  :=  60^1  =  e^e^    (3)  real  quaternions.^ 

234.  Definition.  A.  Dickson  algebra  is  one  whose  coordinates  are  in  an 
abstract  field. 

235.  Theorem.  The  only  Dickson  algebras  (associative)  which  admit  of 
division  are  those  whose  coordinates  are  in  the  Galois  (abstract)  field,  and 
whose  qualitative  units  are  real  quaternions  or  sub-algebras  of  quatemions.- 

■Frobenius  1;  C.  S.  Pkirce  4;  Cartan  3;  Grissemakn  1. 
s  Weddebbcrn  4.     See  also  Dickson  6  and  7. 


60  SYNOPSIS  OF  LINEAR  ASSOCIATIVE  ALGEBRA 

X.  NUMBER  THEORY  OF  ALGEBRAS. 

236.  Definition.  The  number  theory  of  an  algebra  is  the  theory  of  domains 
of  numbers  belonging  to  that  algebra.  Algebras  usually  do  not  admit  of 
division,  unambiguously,  hence  the  term  domain  is  taken  here  to  mean  an 
ensemble  of  numbers  such  that  the  addition,  subtraction,  or  multiplication  of 
any  of  the  ensemble  give  a  result  belonging  to  the  ensemble.  The  first  case 
which  has  been  studied  is  that  of  quaternions,  which  admits  division.' 

237.  Definitions.  An  infinite  system  of  quaternions  is  a  corpus  if  in  this 
system  addition,  subtraction,  multiplication,  and  division  (except  by  0)  are 
determinate  uniquely. 

A  permutation  of  the  corpus  is  given  by  \f^^\  if  through  the  application 
of  this  substitution,  every  equation  between  quaternions  in  the  corpus 
remains  an  equation.     Hence 

f{a  +  b)  =  f{a)+f{b)  f(ab)=f{a).f{b) 

If  n  is  the  corpus  of  all  quaternions  we  have  the  substitutions 

(2)    /(a)  =  flo  =t  «i  K  ±  «2  h  =^  ^^S  ")  i^>  y  is  a  permutation  of  the 
indices  1,  2,  3. 

238.  Theorem.  If  R  is  the  corpus  of  rational  quaternions,  then  a  is 
rational  when  Ug,  aj,  a,,  a^  are  rational.  The  permutations  for  the  rational 
corpus  ai*e  q  (  )  q"^,  and  (a,  /?,  y). 

239.  Definitions.  If  p  =  5  (1  +  i'l  +  ig  +  is),  and  (?  =  A'op  +  ^i  rj  +  A-,  ij 
+  A-gig,  where  Ic^,  Ici,  ho,  k^  are  any  integers,  q  is  said  to  belong  to  the  integral 
domain  J. 

If  the  cooi  Jinates  of  q,  h  A-q,  J  A^  +  A-j,  i  A.-^  +  k^,  i  A;o  +  A;3  are  integers, 
q  belongs  to  the  sub-domain  Jo* 

An  integral  quaternion  is  one  which  belongs  to  /. 

An  integral  quaternion  a  is  pre-divisible  (^post-divisible)  by  h  if  a  z=  be 
(a  =  cb)  for  some  integral  quaternion  c. 

If  e  and  £~' are  both  integral,  e  is  a  umV.     It  follows  tliat 

N{E)={TEf=   1 

There  are  24  units: 

.     ±  1  ±  i,  zh  i»  ±  t 
f=±l,   ±ii,   ±  *2>   =t  *3, -^ 

240.  Theorem.  If  a  =  «c,  then  a^=^  c^v  only  if  y  =  re  or  ?v,  where 
^  =  1  -}-  i|,  and  r  is  any  real  integer. 

'HuRWiTZ  1.     Of.  LirscniTZ  2.     The  first  reference  applies  to  all  sections  following  to  §257. 


NUMBER  THBORY  OF  ALGEBRAS  61 

241.  Theorem.    I  fa  and  h  iiru  intcgriii  (|iiiit,eriiioMK, // :^  0,  we  can  find 
q,  f,  Y, ,  f,  so  that 

a  =  qb-\-c  a  =  hq,  +  6-j  N{c)  <  N{h)  N{r.,)  <  N{h) 

242.  Theorem.    Every  two  integral   (|uuternioMs  a  and  />,  which  are  not 
both  zero,  have  a  highest  common  post-factor  of  the  form 

d  =  (ja  +  hb  ((J J  hj  integers) 

and  a  highest  common  pre-factor  of  the  form 

di  =  or/,  4-  A//,  (^,,  /<, ,  integers) 

243.  Theorem.    The  quaternions   0,   I,  (j,  f  form   a  complete  system  of 
residues  modulo  ^. 

244.  Theorem.    A  quaternion  belongs  to  J^  if  it  is  congruent  to  zero  or  1 
(mod  0- 

245.  Theorem,    li  N  .  a{=  N  .  K  a)  \s  divisible  by  2',  then  a  =  '('h  where 
h  has  an  odd  norm. 

246.  Theorem.    The    following    quaternions  form    a    complete   system   of 
residues,  modulo  2 : 

1,  h,  H,  «3  '-^ 0,   1  +  ij,    1  +  I.,,    1  +  i, 

247.  Definition.    A  primari/  quaternion  is  one  which  is  congruent  to  either 
1  or  1  +  2p  (mod  2^).     Every  primary  quaternion  belongs  to  Jq. 

Two  integral  quaternions  are  ^)re-  (post-)  associated,  if  they  differ  only 
by  a  pre-  (post-)  factor  which  is  a  unit. 

248.  Theorem,    Of  the  24  quaternions  associated  to  an  odd  quaternion  i, 
only  one  is  primary. 

249.  Theorem.    The  product  of  two  primary  quaternions  is  primary. 

250.  Theorem.    If  b  is  primary  then  when 

(1)  b=l  (mod  2^),  K.  b  is  primary, 

(2)  b  =  l+2p  (mod  2^),  —K.  b  is  primary. 

251.  Theorem.    If  ??i  is  a  positive  odd  number,  the  m*  quaternions 

?o  +  ?i h  +  q-z h  +  qsh  (qo,  qu  qi,q^  —  ^,  i,  2 . . . .  m  —  1) 

form  a  complete  system  of  residues  modulo  m. 

These  quaternions  are  holoedrically  isomorphic  with   the  linear  homo- 
geneous integral  binary  substitutions : 

ccj  =  aa-'i  +  /3a-2  xl  =  yxi  +  6x0  (mod  m) 

N{aS  —  (3y)  =  N.q  {mod  m) 

252.  Theorem.    The  number  of  solutions  of  N{q)  =  0  (mod  m),  q  being 
prime  to  m,  is  vi^U  (l  —     .,  K^  H ) 


62  SYNOPSIS  OF  LINEAR  ASSOCIATIVE  ALGEBRA 

The  number  of  solutions  of  N  {g)=l  (mod  m)  is  m^Tlfl 2}' 

These  form  a  group  G,n  which  is  holoedrically  isomorphic  to  the  group  of  the 
linear  homogeneous  binary  unimodular  integral  substitutions,  modulo  m. 

253.  Definition.    7t  is  a  ^n'me  quaternion  when  its  norm  is  prime. 

254.  Theorem.  There  are  ^j  +  1  primary  prime  quaternions  whose  norm 
equals  the  odd  prime  p. 

255.  Theorem.  If  N.  c=p''q'' ....  then  c  =  Ti^Tta ....  ni,Xi ....  x^. ... .  where 
Ttj,  X, . .  .  .  are  primary  prime  quaternions  of  norms  p,  q,  etc. 

256.  Theorem.  If  m  is  any  odd  number,  there  are  ^  (m)  =  2  .  (^  (sum  of  the 
divisors  of  m)  primary  quaternions  whose  norm  equals  m. 

257.  Theorem.  The  integral  substitutions  of  positive  determinant  which 
transform  xl  +  sq -\-  x:^ -\-  xl  into  a  multiple  of  itself  are  given  by  the  equations 

1/  =  ax(3  y  =  —  a  .  Kx .  /B 

where  a,  (3  are  any  two  integral  quaternions  which  satisfy  the  conditions 
a/?  =  0  or  1  (mod^. 

258.  Definition.  The  general  number  theory  of  quadi-ates  has  been  studied 
recently  by  Do  Pasquier.^  A  number  in  a  quadrate  algebra  he  calls  a  feftarion. 
It  is  practically  a  (square)  matrix  or  a  linear  homogeneous  substitution.  An 
infinite  system  of  tettarions  is  a  corpus,  if  when  a  and  ^  belong  to  the  system, 
a  ±  ^,  a  .  ^,  (3  .  a,  a  :  (3,  (S~^.a  belong  equally  to  the  system.  A  substitution 
of  a  tettarion  t  =  /(t)  for  a  tettarion  t  is  indicated  by  [t,  /(t)].  K  permuta- 
tion is  a  substitution  such  that  when  a  is  derived  from  n  tettarions  aj. .  •  -a,,  by 
any  set  of  rational  operations,  so  that  a  =-fi(ai.  .  •  .a„),  then /(a)  =  a  is 
derived  from  aj.  . .  .a„  by  the  same  set  of  rational  operations,  so  that 

a  =  .B  (cci .  .  .  .  a„) 

259.  Theorem.  The  substitution  [a,  /"(a)]  is  a  permutation  of  the  corpus 
\K\,  if  the  tettarions /(a)  do  not  all  vanish,  and  if 

/(a  +  /3)  =/(a)  +/(/?)  f{a[3)  =  /(a)  /(^) 

for  any  two  tettarions  a,  (3  in   \K\.     The  tettarions  /(a)  also  constitute  a 
cori)us. 

260.  Definition.  An  inversion  of  the  corpus  is  a  substitution  such  that  not 
all  /(t)  are  zero,  and  also  for  any  two  tettarions  a,  (3  we  have 

f{a  +  (3)  =  f{a)  +  /(/?)  /(a/3)  =  /{(3)  /(a) 

[t,  t]  is  an  inversion,  where  t  is  the  transpose  of  t.     If  [«,/(«)]  is  the  most 
general  substitution  of  the  corpus,  [a, /(a)]  is  the  most  general  inversion. 

261.  Definition.    Two  permutations  of  the  form 

[a,  /(a)]  and  [a,  q .  /(a) .  q-'} 
where  q  is  any  tettarion  which  has  no  zero-roots,  are  said  to  be  equivalent.    All 
equivalent  permutations  constitute  a  class. 

>  Do  Pasquiku  1.     This  reference  appHes  to  SS 858-297. 


NUMBRR  THEORY  OF  ALGEBRAS  g3 

262.  Theorem.  The  .siib.stitution  t,,t  '  is  a  perniutation  of  the  corpun  o 
of  all  tettario.is  of  order  .v;  where  t  is  a  tettarion  such  that  Nit)^:  0  N  U) 
being  the  .s-th  or  last  scalar  coefficient  in  the  characteristic  equation  of 't  The 
coefficient  N{t)  is  called  the  norm  o{  t. 

263.  Definitions.  A  tettarion  is  m^ioW  if  all  its  s^  coordinates  are  rational 
All  rational  teltarions  form  a  corp„s  \R\.  All  tettarions  whose  coordinates 
belong  to  a  given  domain  of  rationality  constitute  likewise  a  corpus  A 
rational  tettanon  is  integral  if  all  its  coordinates  are  rational  integers. 

The  integral  tettarion  a  is  pre-  (or  post-)  divisible  by  the  tettarion  3  if 
an  integral  tettarion  y  can  be  found  such  that  a  =  f3y  (or  a  =  y^)  A  unit 
tettarion  .  is  an  integral  tettarion  which  is  pre-  (post-)  divisible  into  every 
integral  tettarion.  When  Nie)  =  +  1  we  call  .  a  proper  unit-tettarion ;  when 
-^  (^) 1  we  call  e  an  improper  unit-tettarion. 

264.    Theorem.    Let  a,.  =h  +  e„  where  h  is  the  modnhis  of  the  quadrate 
that  IS,  IS  scalar  unity,  and  e^.  is  one  of  the  s^  units  defining  the  quadrate ;  and 

^^^  ^  ~   5   *'' '''  ^^  *"^  integral  tettarion  ;  then  among  the  tettarions 


'^     ~o.ur  (a;=l,  2....) 


there  is  always  one  such  that  a  certain  pre-assigned  coordinate,  say  tif  is  not 
negative,  and  is  less  than  the  absolute  magnitude  of  any  other  coordinate  of  r 
of  the  form  t,j  {k=l....  s,  k  zfz  i),  provided  t,j  :^  0. 

265.  Definition.  A  tettarion  yf^e,  in  which  all  coordinates  for  which 
^  >  A;  (or  *  <  ^  vanish,  ,s  said  to  be  pre-  {post-)  reduced.  They  constitute  a 
sub-corpus^  lettarions  of  the  form  y,t,,e,  are  both  pre-reduced  and  post- 
reduced  The  components  ^,(.  =  1 .....)  in  a  reduced  tettarionr  vanish  only 
when  T  has  zero-roots.  ^ 

266.  Theorem.  If  t  is  any  integral  tettarion,  a  proper  unit-tettarion  e 
may  be  found  such  that  .  .  r  (or  r  .  ^)  is  a  pre-  (post-)  reduced  tettarion,  in 
which,  of  all  the  coordinates  <«,  at  most  only  t^  can  be  negative.  This  co- 
ordinate is  negative  only  if  iV(T)  <  0. 

If  r  is  any  integral  tettarion,  we  may  find  a  pair  of  proper  unit-tettarions 
e,  and  e,  such  that  e,  re,  is  of  the  form  2  cZ,  .„  (i  =  1  . .  . .  ,),  and  among  the 
coordinates  at  most  only  <,  is  negative,  and  cZ.,  is  divisible  by  d-  ,  ._,  The 
coordinate  d,,  is  negative  only  when  A^(t)  is  negative.'  '      ' 

If  a  =  ei  /?f2,  a  and  /?  are  said  to  be  equivalent. 

267.  Theorem.  Every  proper  unit-tettarion  e  is  expressible  in  an  infinite 
number  of  ways  as  the  product  of  integral  powers  of  at  most  three  unit- 
tettarions.     These  three  may  be 

tty  =  A  -f-  ey 

^ij  =  ^  e,^  +  Cij  -  eji  {k=i  ....  s,  k  4:  i,  k  :^j) 
7  =  ^21  +  %  +  . . . .  +  e,,_i  —  e„ 

'Cf.  Kkoneckbk:   CrelleW,  135-136;   Bachman:  ZahlenthtorU  IV  T^il,  29i. 


64  SYNOPSIS  OF  LINEAR  ASSOCIATIVE  ALGEBRA 

268.  Theorem.  Every  integral  tettarion  r  is  equivalent  to  a  tettarion  of 
the  form  S  tu  e^.  The  coordinates  less  t  are  the  shear  factors  of  the  character- 
istic equation  of  T.  The  norm  of  t,  N'{r),  is  the  pr9duct  of  these  coordinates. 
Two  tettarions  are  equivalent  when  they  have  the  same  shear  factors  and  the 
same  nullity. 

269.  Theorem.  In  order  that  a,  be  a  factor  of  t  =  aj  .  .  . .  a^  . . . .  a,  it  is 
necessary  and  sutficient  that  the  nullity  of  ai  be  not  higher  than  that  of  t,  and 
that  each  shear  factor  of  a,,  or  combination  of  shear  factors,  be  divisible  into 
the  corresponding  shear  factors  of  t.  If  an  integral  tettarion  t  is  a  pi'oduct 
of  others,  then  every  combination  of  shear  factors  of  t  is  divisible  by  the 
corresponding  combination  of  shear  factors  of  any  one  of  these  others. 

270.  Definition.  Two  tettarions  t  and  er  are  called  pre-associated .  The 
association  is,  proper  or  impropter  according  as  N{£)  =  +  1  or  —  1.  Associated 
tettarions  form  a  class.  The  simplest  representative  of  a  class  will  be  called 
a  primary  tettarion. 

A  pre-primary  tettarion  p  =  ^Pijeu  satisfies  the  following  conditions  : 

Pij  =  ^     i>J         Pn  =  ^  and  Otpij<Pji 
for  all  i<^j  and  if  pjj  -^  0;  i  and  j  have  values   1  .  .  .  .  s  subject  to  the  con- 
ditions stated. 

271.  Theorem.  A  primary  tettarion  cannot  have  a  negative  norm.  Pri- 
mary tettarions  with  zero-roots  are  infinite  in  number,  but  the  number  of 
primary  tettarions  of  a  given  norm  m  :|:  0  is  finite. 

272.  Theorem.    If  wi  =  n  j:5f   ,  where  p^  is  a  prime  number,  and  if  xijn) 

i 

is  the  number  of  distinct  pre-primary  tettarions  of  norm  m,  then 

xM  =  xipf)  x(pf')  ■■■■xipf) 

_  (ff«  +  I-l)(p"+2_i)    ....(^,a-f.-l_l) 
nP)-  (^,_l)Q/_l)....(^s-l_l) 

273.  Theorem.  If  r  is  any  integral  tettarion  and  m  is  any  integer  (rf:  0) 
then  an  integral  tettarion  a  may  be  found  such  that  r:=ma  or  else  T=^7nG  +  a 
wherein  a  is  an  integral  tettarion  and  0  ■<  |  N{a)  ]  <!  |  m'  | . 

274.  Tlieorem.  If  a  and  8  are  two  integral  tettarions  of  which  8  has  not 
zero-roots,  then  two  integral  tettarions  j3  and  y  may  always  be  determined 
such  that  either 

a  =  (3h    and    y  =  0    or   a  =  {38  +  y    and    0  <  |  A^(y)  |<  |  A^(5)  | 

By  this  theorem  a  highest  common  pre-  (post-)  divisor  ma}'  be  found  by 
the  Euclidean  method  for  any  two  integral  tettarions. 

275.  Definition.  An  infinite  system  of  tettarions  which  do  not  all  vanish 
is  a  pre-  (jjost-)  ideal  if  when  Tj  and  Tj  belong  to  the  system,  ^Tj,  Tj  ±  Tj  also 
belong  to  the  system,  where  y  is  any  integral  tettarion. 


NUMBER  THEORY  OF  ALGEBRAS  66 

276.  Theorem.  If  rj  .  .  .  .  t„  are  iiitej^ral  lettarions,  which  do  not  all 
vani.sii,  tljcii  tlio  totality  of  tettarions  7,  t,  +  ....  y„r„  wliere  yj  .  .  .  .  y„  run 
independently  throiij^h  the  range  of  all  integral  tettarions,  is  a  posl-ideal. 
The  tettarions  t,  ....  t„  are  said  to  form  the  basis.  An  ideal  with  one 
tettarion  in  its  ba.sis  is  a  principul  ideal.  It  is  designated  hy  {ry)  or  (yr). 
Two  ideals  are  equal  if  they  contain  the  same  tettarions. 

277.  Theorem.  Pre-  (post-)  associated  tettarions  generate  the  same  prin- 
cipal post-  (pro-)  ideal.  If  two  integral  tettarions  without  zero-roots  generate 
the  same  principal  post-  (pre-)  ideal  they  are  pre-  (post-)  associated. 

278.  Theorem.  Every  pre-  (post-)  ideal  generated  by  rational  integral 
tettarions  which  do  not  all  have  zero-roots  is  a  principal  ideal. 

279.  Definition.    Nul-idenls  contain  only  tettarions  with  zero-roots. 

280.  Theorem.  An  ideal  which  is  both  pre-ideal  and  post-ideal  cannot  be 
a  nul-ideal. 

281.  Theorem.  Every  n  given  integral  tettarions  a,  (3  . .  ■  ■  (i  which  do  not 
all  have  zero-roots  possess  a  highest  common  divisor  h  which  is  expressible  in 
the  form 

5  =  a^i  +  /?^2  +  •  •  •  ■  +  i"^.i    or    h  =■  hi(x  +  ^o^  +....-{•  Sn(i 
wherein  5j(i=  1  ....  n)  are  definite  integral  tettarions.     Every  pre-  (post-) 
divisor  of  5  is  a  common  divisor  of  a  ....  jU  and  conversely.     Moreover  S  is 
determined  to  a  factor  which  is  a  post-  (pre-)  unit-tettarion. 

282.  Theorem.  If  rx  and  /3  have  no  common  pre-  (post-)  factor,  then  two 
tettarions  y,  6  may  be  found  such  that 

ay  +  ^6=1    or    ya  +  6(3  =  1 
If  a  and  (3  have  a  common  factor  these  equations  cannot  be  solved.     If  a  and 
/?  have  a  common  divisor  which  is  not  a  unit-tettarion  then  N{a)  and  N{l3) 
have  a  common  divisor  other  than  unity.     The  converse  of  this  theorem  and 
the  theorem  imply  each  other  if  one  of  the  tettarions  is  real,  that  is,  of  the 

s 

form  a  2  e^j. 
<=i 

283.  Definition.  An  integral  tettarion  which  is  not  a  unit-tettarion  nor  has 
zero-roots  is  prime  if  it  cannot  be  expressed  as  the  product  of  two  integral 
tettarions  neither  of  which  is  a  unit  tettarion. 

284.  Theorem.  The  necessary  and  suflBcient  condition  that  7t  is  a  prime 
tettarion   is  that  its   norm   N  [n)  is  a  rational   prime  number.      There  are 

»'  —  1    . 

— ^~:  different  primary  primes  of  norm  p. 

285.  Theorem.  If  h  is  an  integral  tettarion  of  the  form  2  </,,  e,j  and  if 
N {h)  =■  p'^ q'' .  .  .  .t",  where  p,  q.  .  ■  -t  are  distinct  primes,  not  including  unity, 
then  S  can  be  factored  into  the  form 

,S  =  Ttj  ....  7t„  Xi ....  Xb  ....  Ti  ..  .     T„ 


66  SYNOPSIS  OF  LINEAR  ASSOCIATIVE  ALGEBRA 

where  n^.  .  ■  -rCa  are  primary  prime  tettarions  of  norm p,  xi-  .  ■  -x^  are  primary 
prime  tettarions  of  norm  q,.  .  .  .r^.  . .  -Tn  are  primary  prime  tettarions  of  norm 
t,  all  being  of  the  form 

Xpueu 2<«ete 

286.  Definition.  An  integral  tettarion  is  primitive  if  its  coordinates  have  no 
common  divisor  other  than  unity.  It  is  primitive  to  an  integer  m  when  its 
coordinates  are  all  prime  to  m. 

Every  primary  tettarion  is  also  primitive. 

287.  Theorem.    Let  y  be  a  primitive  integral  tettarion  and 

where 2?,  q-  ■  •  -t  are  the  prime  factors  o^ N{y).  Then  y  can  be  decomposed  in 
only  one  way  into  the  form 

where  e  is  a  unit  tettarion,  and  Tti- .  ■ -Tia,  are  prime  tettarions  of  norm  p, 
Xi- .  •  .Xa„  are  prime  tettarions  of  norm  5,  •  •  •  •  Tj  .  .  •  . T„„  are  prime  tettarions  of 
norm  t.     The  product  of  each  I  successive  factors  is  primary,  where 

Z=  !....«.   (i=  l....n) 

288.  Definition.  If  aj,  a^.  .  . -a^  are  s  integral  tettarions  of  equal  norms 
N{ai)  =  iV^(a2) .  .  .  .  ^  ^{o-s),  and  if  ai  a, .  .  .  .  a^  =  iV(ai),  then  these  tettarions 
are  semi-conjugate. 

289.  Theorem.  A  product  of  any  number  of  prime  tettarions  of  forms 
Sf^iiCu  is  a  primitive  tettarion  of  the  same  form  if  among  the  factors  no  s  of 
them  are  semi-conjugate. 

290.  Theorem.  A  product  of  primary  prime  tettarions  ni...-7t„,  where 
N(^'7ti)=pi  (i  =  I .  .  .  .71)  pi  being  distinct  primes,  is  always  a  primitive 
tettarion. 

291.  Definition.  Two  given  tettarions  a  and  (i  are  pre-  [post-)  congruent  to 
a  moduluti  y,  if  their  difference  a  —  |S  is  pre-  (post-)  divisible  by  y.  This  con- 
gruence is  indicated  by 

a  =  /?  (mod  y,  pre) 

or 

a  —  ^  (mod  y,  post) 

There  is  then  an  integral  tettarion  ^  such  that 

a  —  (3  =  y^  or  '(y 

292.  Theorem.  If  a  and  /?  are  pre-  (post-)  congruent  modulo  y,  they  are 
also  pre-  (post-)  congruent  for  any  tettarion  post-  (pre-)  associated  with  y,  as 
modulus. 


NUMBER  THEORY  OF  ALGEBRAS  67 

293.  Theorems.    If    rx  =  'r     /^  =  t  then  a  = /?  (mod  y) 

11"     a  =  T     ii  =  a  then  arh/S^Tia  (mod  yj 

If    a  =  /3  then  ra  =  r^  (mod  y,  post) 

If    a  =  /?  then  ar  =  (3r  (mod  y,    pre) 

If7<^  =  i^y  a  =  /3  then  a^S/^i^  (mod  y,  post) 

If    rx  E^  /:;      0  =  (5   y(3  =  ^y  then  e«  =  ^/5  (mod  y,  post) 

294.  Definition.  Two  tettarions  a,  (3  are  congruent  as  to  a  rational  integer 
in  ipO,  when  a  —  /?  is  divisible  by  tti,  indicated  by  a  =  ^  (mod  w).  In  this 
case  for  each  pair  of  coordinates  we  have  afj  =  bij  (mod  m).  A  complete 
system  of  residues  consists  of  wi"'  tettarions,  obtained  by  setting  each  coordinate 
independently  equal  to  each  one  of  a  complete  set  of  residues  modulo  m. 

295.  Theorems.  A  tettarion  congruence  modulo  m,  a  rational  integer  can 
be  divided  by  an  integral  tettarion  ^,  without  altering  the  modulus  only  if 
iV(^)  is  prime  to  m. 

A  sufficient  condition  for  the  solubility  of  the  congruence  a^  =  /5  (mod  m) 
by  an  integral  tettarion  ^  is  that  N{a)  is  prime  to  m.  If  this  condition  is  ful- 
filled the  congruence  possesses  one  and  only  one  solution  ^  (mod  m),  namely, 

^  =  r — ^/3  (mod  m),  where  r  is  a  root  of  r  .  N{a)  =  1  (mod  m). 

296.  Definitions.    A  tettarion  with  zero  roots  and  nullity  s  —  r  is  pseudo- 

r 

real  if  it  is  of  the  form  c?jj  "S,  e^^,  r  <is.    A  tettarion  with  zero-roots,  of  the  form 

2  t^jCy  where  t^^  =0  for/ =  r  +  1  •  •  •  -s,  is  singular  or  non-singular  according 
as  its  rank  is  less  than  or  equal  to  r.  The  product  of  the  first  r  latent  roots 
of  a  tettarion  of  this  kind  is  called  its  pseudo-norm  N'.  A  tettarion  of  the  type 
2  <ii  Cji  is  never  singular.     When  a  tettarion  is  singular  its  pseudo-norm  is  zero. 

297.  Theorem.  If  a  and  fi  are  two  integral  tettarions  in  which  coordinates 
of  the  form  (,j  =.  0  for  J  =  r  -{•  1 .  .  .  .  s,  and  if  a  is  pre-reduced,  u  is  pseudo- 
real,  and  ^  0,  then  two  tettarions  of  the  same  type  ^  and  >;,  may  always  be 
found  such  that  either 

a  =  (I  .^  )7  =  0 

where  the  pseudo-norm  of  a  satisfies  the  conditions 

0<\N'{a)\<\N'{f^)\  =  \{m,,y\ 
If  r  and  (3  are  two  integral  pre-reduced  tettarions  of  the  same  type  as 
a,  fi  above,  and  ^  is  non-singular,  then  there  are  always  two  other  pre-reduced 
tettarions  of  this  type  ^  and  y;  such  that 

r  =  5,5  )7  =  0 

^^  r  =  q^  +  y!    and    0 <  | iV^' (>?) [<  1  iV' (,5)  | 

Every  non-singular  post-ideal  based  on  tettarions  of  this  type  is  a  princi- 
pal ideal. 


68  SYNOPSIS  OF  LINEAR  ASSOCIATIVE  ALGEBRA 

XI.     FUNCTION  THEORY  OF  ALGEBRAS. 

298.  Definition.    In  §58,  chapter  II,  we  have  for  any  analytic  function  of  ^, 

This  definition  gives  a  complete  theory,  if  the  roots  may  be  treated  as  known. 
Other  definitions  are  given  below.^ 

CO 

299.  Definition.    2  a,  ^''  defines  an  analytic  function  of  ^,  if  the  roots  of  the 

1 

CO 

characteristic  equation  of  ^  converge  in  the  circle  defined  by  2  a^z",  where  sis 

1 
an  ordinary  complex  number. 

Sa^^"  defines  a  function  of  ^,  if  ^~^  exists,  and  if  the  roots  of  the  char- 

CO 

CO  CO 

acteristic  equation  of  ^  converge  in  the  circles^  of  2  a^z"  and  2  a_^2"'^ 

1  1 

300.  Definition.    Let 

f=^eji{x^ a:,)  i—1 r 

and  let 

(ia;  =  2  Cj  dxi 
Then 

df  =  y^  1^  .  dx,e,  =  f'.dx  =  dz.<f 
If  tt'  =^1,  then /is  an  analytic  function^  v/hen/'  .y  ^  y .'/. 

301.  Theorem.  The  algebra  must  contain  for  every  number  m,  a  number 
V  such  that  ?/_y  =  yv  for  every  y. 

302.  Theorem.  In  a  commutative  algebra  the  necessary  and  sufficient 
condition  of  analytic  functions  is 

303.  Theorem.  The  derivative  of  an  analytic  function  is  an  analytic  func- 
tion. If  two  analytic  functions  have  the  same  derivative,  they  differ  only  by 
a  constant. 

304.  Theorem.  An  analytic  function  is  a  differential  coefficient,  only  when 
the  algebra  is  associative,  distributive,  and  counnutative. 

Ill 

305.  Theorem.    Km  and  v  are  analytic,  uv  and        are  analytic. 

'  Fbobbnius  1 ;    BuoHHEtMT;   Sylvester  4  ;    Tabbk  C,  7.  'Wetr  7. 

'ScHEFFEKS  8.     Applies  to  §§300-306. 


FtJNCTION  THEORY  OF  ALGEBRAS  69 

306.  Theorem.    ir/(px)  =  f(x),  tlicn  /(y)  is  a  constant. 

307.  Theorem.  For  a  Weierstuass  commutative  algebra,  let  «,,  h,  he  in 
the  i-th  elemcutary  algebra,  and 

a  =  2  cit         b  =z2hi 

Then  a  ±  i  =  2  (a,  ±  b,)     ab  =  Xa, hi       "   =  2  ^'  ,  if  b  is  not  a  nilfactor. 

If  hi  =  0,   for  t  =  1 . .  .  .  ij,  6t  ^  0  i>  ii 

and  if  a^  =  0,   for  t  :=  1 .  .  .  .  ij 

then  -^  =  2  -T-  i  >  M 

In  any  other  case  the  division  of  a  by  Z/  gives  an  infinity} 

308.  Theorem.  The  sum,  difference,  product,  and  quotient  of  two  poly- 
nomials is  formed  as  in  ordinary  algebra. 

309.  Theorem.  The  number  of  solutions  of  an  algebraic  equation  of  degree 
p  is  N-=.  p^',  when  each  elementary  algebra  is  of  order  two. 

If  ri  of  the  elementary  algebras  are  of  order  one,  and  ?•• — r^  of  order 
two,  N=-p\ 

In  any  case  the  number  of  infinities  and  roots  is  ^/.  The  nun)ber  of  roots 
is  infinite  if,  and  only  if,  the  coefficients  are  inultijiles  of  the  same  niliaclor.- 

310.  Theorem.  A  polynomial  i^(^)  can  not  vanish  for  every  value  of  ^ 
unless  its  coefficients  all  vanish. 

Two  polynomials  equal  to  each  other  for  every  value  of  ^,  must  have 
the  coefficients  of  like  powers  of  ^  equal. 

311.  Theorem.  If  an  algebraic  polynomial  i^(^)  is  divided  by  ^  —  ^',  ^' 
being  a  root,  the  degree  is  reduced  to  {p —  1)  and/j*""  —  {p —  l)'""  roots  have 
been  removed. 

In  ordinary  complex  algebra  r=  2,    jj**"  —  {p —  l)'""  =  1. 

312.  Theorem.  If  two  polynomials  have  a  common  root,  ^',  they  have  a 
common  divisor  ^  —  t,'. 

313.  Theorem.  If  F{j^)  is  differentiated  as  if  ^  were  an  ordinary  quantity, 
giving  F'  {Q,  then  the  necessary  and  sufficient  condition  that  there  is  a  system 
of  roots  of  i^(^),  having  just  p  equal  roots,  is  that  F'  (^  has  at  least  one 
system  of  roots  of  which  p — 1  are  this  same  equal  root,  and  that  no  system  of 
roots  of  F'{(^)  has  this  root  more  than  p — 1  times.  F{^)  and  F' {^)  have 
therefore  the  common  divisor  (^  —  ^'Y'^' 

314.  Theorem.  It  is  not  always  possible  to  break  u-p  ,,  ^L  into  partial  frac- 
tions. 

'Bbrlott  1.     Applies  to  §§307-315.  'Weierstbass  2. 

6 


70  SYNOPSIS  OF  LINEAR  ASSOCIATIVE  ALGEBRA 

315.  Theorem.  If  ^  is  considered  to  be  written  in  the  form  ^  =  22;  Xj, 
where  i  =  1  . .  . .  rj,  and  Zj  is  any  real  or  complex  number,  the  whole  theory  of 
functions  of  a  comj^lex  variable  may  be  extended  to  numbers  which  are  not 
nilfactors.  If  there  are  nilfactors,  meromorphic  functions  must  be  treated 
specially.     We  have 

316.  The  treatment  of  quaternion  and  biquaternion  differentials,  integrals, 
and  functions  may  be  found  in  the  treatises  on  these  subjects  and  references 
there  given  ;  references  are  also  given  at  the  end  of  this  memoir.  The  general 
principles  of  such  forms  may  easily  be  extended  to  any  algebra.  Differentia- 
tion and  integration  along  a  line,  over  a  surface,  etc.,  may  also  be  found  in 
the  appropriate  treatises. 

The  problem  of  extending  monogeneity  to  functions  of  numbers  in 
quadrate  algebras  has  been  handled  recently  by  Autonne.^  His  results  are 
as  follows  : 

Let  ^  be  any  number  in  an  algebra,  and  let  H  be  a  number  wliose 
coordinates  are  functions  of  those  of  ^.     The  index  of  monoyeneity  N  is  the 

N 

minimum  number  of  terms  necessary  to  write  cZH  in  the  form  2  Gi  .  dt,  .  r^,. 

i  =  \ 

'9 
wherein  Ci  and  Tj  are  functions  of  ^.     If  we  write  v  =  2  e{  >,  -   ,  we  have  in  all 

i  =  \       VX^ 

cases  c^H^ /•  (^^V  •  H  =  T(t7$).     The  Jacobian   of  the  coordinates  of  E   is 
then  7n,.(T). 

1  —  r 

If  now  we  put  T=  2  u^j  K^i,  where  K^^i  =  e,,  Q  e, ,  we  may  find  the 

H 

scalars  w^i  uniquely  if  the  algebra  is  a  quadrate."     For,  indicating  quadrate 
units  by  a  double  sufiBx,  and  writing  n"  =  r, 

ijkl 

and  if  we  operate  on  Cj^.  and  take  /.  e^  ()  over  the  result, 

l....r 

If  we  put  'P  =  2  ■?%  .  Ci  /e,  (),  or  in  the  case  of  a  quadrate, 

hi 

\....n 

*  =  2  w^j„i .  ey  lea  0 

ijkl 

then  the  rank,  that  is,  ?i — v,  where  r  is  the  nullity  of  ip,  is  the  index  of 
monogeneity,  N.     N  is  invariant  for  a  change  of  basis. 

The  transverse  of 'P  corresponds  to  interchanging  cr,  and  r^.     For 


and  /e^y  Ten  =  w, 


„  1 n  1 — n 

*  =  2  lea  ^ejk  ■  ^ki  l^a  —  2  le^j  Ten  .  e^  /e„ 

ijkl  ijkl 


'  AUTOMNB  5,  6.  '  HACSDOBFr  1. 


FUNCTION  THEORY  OP  ALGEBRAS  7  j 

Let  P  0  =  2  Ci  me,  0  0,  ti.o  ^^  forming  a  ^-pair.     Then  P  =  P,  and  we 


i  ^1 
have 


?r  =  :S.w,,^e,me,c,{)e,^) 


kl 


=  2  rvki  2  c,  hi  0  .  /^  (e<  Cj  c,  e^t  0 
PT  =  2«;«2e,/e,()./^(e,eie,e,0 

Hence   PT  =  T  P  =  PT    if    ip  = 'P,    and    conversely.       Again    PT^PH/y, 
therefore  ifPT  =  fP  we  have 

PH/v  =  y/P3 
Operating  on  cZ^,  we  Iiave 

c? .  PH  =  ^Td^PB  —  \7lBdP^ 

Hence  if  y;  =  P^,  /&?>?  is  an  exact  differential.     Thus  if  V  is  self-transverse, 
/E(/>7  is  an  exact  differential  and  conversely. 

When  N=  1,  we  have  H  in  one  of  the  four  following  types : 

I.     B  =  K^A  +  M  (K,  A,  M,  constant) 

II.     H  =  2  Z  (^e,i)  fi,i  (Z  arbitrary) 

i^  1 

III.  E  =  2  ^n  Z  (fii,^)  (Z  arbitrary) 

IV.  B="i\ufr,{t)p,{t)dt 


tj 


<  =  t|'  [/.  a  («'u  ^)  .  ■  .  •  I.  a   (e,„i^)],   and   i^,  >7,-,  p^   are    arbitrary    scalar 
functions  oi  t;  a  is  any  constant  number. 


72  SYNOPSIS  OP  LINEAR  ASSOCIATIVE  ALGEBRA 

Xn.     GROUP  THEORY  OF  ALGEBRAS. 

317.  This  part  of  the  subject  is  practically  undeveloped,  although  certain 
results  in  groups  are  at  once  transferable  to  algebras.  A  considerable  body 
of  theorems  may  thus  be  got  together,  esjDecially  for  the  quadrates.  For 
example,  the  groups  of  binary  linear  homogeneous  substitutions  lead  at  once 
to  quaternion  gi'oups,  ternary  linear  homogeneous  substitutions  to  nonion 
groups,  etc.     It    is  to   be  hoped  that  this  branch  may  be  soon  completed.^ 

318.  Definition.  A  group  of  quaternions  is  a  set  of  quaternions  Ji  •  •  •  ■  ?„ , 
such  that 

qTi  =  l  9[iqj  =  qk  i,j=l....n 

TTii  is  a  positive  integer,  and  k  has  any  value  1,  2....n.  The  quaternions 
give  real,  complex,  or  congruence  groups  according  as  the  coordinates  are  real, 
complex,  or  in  an  abstract  field. 

319.  Theorem.  To  every  quaternion  q'=-w-\xi  -\-  yj  +  zlc  corresponds 
the  linear  homogeneous  substitution 


/w -\- z  *^ — 1     — 2/ -j- a;  V — 1\ 


and  conversely.     The  determinant  of  the  substitution  is  I'q.     To  the  jiroduct 
of  two  quaternions  q,  r,  corresponds  the  product  of  the  substitutions. 

320.  Theorem.  To  every  group  of  binary  linear  homogeneous  substitutions 
corresponds  a  quaternion  group,  and  conversely.  To  every  group  of  binary 
linear  fractional  unimodular  substitutions  corresponds  a  group  of  quaternions 
multiply  isomorphic  with  it,  and  to  every  quaternion  group  corresponds  a 
group  of  binary  linear  fractional  unimodular  substitutions,  the  latter  not 
alv?ays  distinct  for  diflerent  quaternion  groups. 

321.  Theorem.  To  every  quaternion  of  tensor  Tq  corresponds  a  Gaussian 
operator  Tq.q  Oq^^  =  Oq,  and  conversely. 

If  2 .  r  =  s,  then  Gg .  (?,.  ^  G^- 

Hence  groups  of  these  Gaussian  operators  are  isomorphic  with  quaternion 
groups,  and  conversely,  but  tlie  isomorphism  is  not  one-to-one. 

322.  Theorem.  To  every  unit  quaternion  q,  there  corresponds  a  rotator 
^,^  =  2  ()  g-i,  and  conversely,  the  same  rotator  corresponding  to  more  than 
one  quaternion. 

Likewise  a  reflector  Ii,j=:  —  qQ  q~^,  and  conversely. 

Further,  for  any  fixed  quaternion  a  admitting  of  a  reciprocal,  there  cor- 
responds the  a-transverse  of  q, 

T\f  =  aqa-^ 

'  Cf.  Ladbbnt  8,  4. 


GUOUP  THEORY  OF  ALGEBRAS  73 

Thus  if  qr  =  8, 

R,^ .  n,  -  R,  R.,  .11,  =  —  R,  Tf^ .  T\  =  T'i") 

Thus  to  every  group  of  quaternions  g-i. . .  .(/„,  corresponds  the  rotator  group 
R^^ ....  R^^ ;  tlie  reflector  group  ±  R,^^ ,  ±  R,^„  .  .  .  .  ±  R,^^ ;  and  the  transverse 
groups  7',*,'''  ....  T;il.  If «  =  1,  the  transverse  group  is  the  group  of  conju- 
gates ;  and  if /Set  =:  0,  we  Iiave  a  group  of  transverses  in  the  matrix  sense. 

323.  Theorem.  If  weconsider  that  gand  —  q  are  to  be  equivalent,  </  =  —  q, 
then  tlie  rotation  groups  give  tlie  quaternion  groups  as  follows: 

To  C„  corresponds  h"n  ,  r  =  1 .    .  .  n 

Z>„  corresponds  lcn,i,  r  =  1 . .  .  -n 

T  corresponds  1,  i,j,  h,  (1  ±  irby  ±  Ic) 
0  corresponds  1,  i^,  fi,  ki,  ^(1  ±  i±j±i  /•) 

hx/2{i±J),  iv/2(y±7.;),   hK/2{fc±i)  r=l,   2,  3 

7  2'!//..       „,  »„ov    7  2h" 

_  17  2/1      ."h    k  6  (i  -{-  2/e  cos  72  )  «  6 

/   corresponds   k  =  ,  jk  5  ,  -A^^_p.^^^._=- 

A;fy(l  +J/^cos  72°)  7cT  ^^   ^^,   ^„  ^  ^  . 

V  1+4008^72°  '     ' 

324.  Theorem.  To  the  extended  polyhedral  groups  correspond  the  follow- 
ing five  quaternion  groups : 

in 

To  Cr  corresponds  the  group  hr ,  of  order  7;   {k  any  unit  vector, 
n=  1  ....  r). 

To  Dl  corresponds  the  group  7c^  i'\  of  order  4r,  {Sik=  0,  i?  ■=■ —  1, 
n=:  1  .  .  .  .r;  7t  =  1  .  .  .  .4). 

To   T  corresponds  the  group  of  order  24:    ±1,    ±  t,    d=  y,    ±  ^, 

i(dz  1  =b  t  ±y  ± /.•). 

To   0'  corresponds  the  group  of  order  48  :    ±  1,   ±  i,   iy,   rfc  h^ 
^  (±  1  ±  ?•  ±  y  ±  A)  ^V2  (±  1  d=  i)  i  V2  (±  1  ±  y) 

i  V2 ( ±  1  ±  /.•)  i>/2  (±  i  ifc  y)       i  V2  (it  y  ±  z,-) 

i  ^2  (  ±  A;  db  i) 
To  /'  corresponds  the  group  of  order'  120  :  ±  k\  ,   ±:jkT, 
it  k''s  (i  +  (o/r)  ks  it  k'i  {i  +  ak)  jkT 

V  1  +  0)^  Vl  -\-  (^ 

•  ■5         n  +  s  =  it  l,it  2(mod5)  u  =  2cos  72°  =  J  (— 1 -f  V5) 

1  Of.  Stbimobah  3. 


where 


74  SYNOPSIS  OF  LINEAR  ASSOCIATIVE  AIXJEBKA 

325.    Theorem.    Combinations  of  rotations  and  reflections  give  the  poly- 
hedral and  the  crystallographic  groups.     Thus  we  have  correspondences  : 

C,    =kT{)Jc--:  n=l....   r 

Dr=kr{)k-J      i{)i-'  n=l....   r 

T  =1   h{)h-'   i{)i-'  j{)r'    {\±i±j±ic){){\±i±j±k)-' 

0    =1     i{)i-^      JOf^     k{)k~^      (1  d=iifcy±  ^)()  (1  ±i±yd=A:)-' 

(1  ±  00(1  ±  i)-'     (1  ±y)()(i  ±y)-^     (1  ±  k)  0  (1  ±  k)-' 

[i  ±  j) 0  ( i  ±y)-i      (y ±  /.•)  0  (y  ±  k)-'      {k  .4=  i)  o  (/.:  ±  i)- 

/     =  Z;^  0  k-i  J ()y-i  (i  +  2  cos  7 2°  /;•)()  (i  +  2  cos  7 2°  k)'^ 


2r 

.    r 

.  2/- 


and  their  combinations. 

6^;  z=  [— k'^ {)  7c-''-^Y  h  = 

(_!'J  =z  k'v  0  /.-.^T  —  /i-  0  A;   '  and  couibination.s  /*  = 

C'J'  =^  k~r  {)  7c~ ~r  —  i{)i~^  find  combinations  h  = 

D'r   =  [ — kH'  0^^'^  ]     ^Oi"^  and  combinations  A  =: 

Z>,'.'  =  ^T  0 &~  7     — k\)k~^   i()t~' and  combinations  A  = 

Z)r"=  k~r  {)k~~f     — a()a~*    iOi"^    oil^  ii^nd  combinations  Ji  = 

rpi      rp  m 

T"  =  T  [ — (* — y)  0  (i — y)"']  a.nd  combinations 

0'  =  0  —  0 

/'  =/  — / 

326.  Theorem.    If  /S'.  e  =  f,  t'"  =1 ;  then  the  product  of  each  group  in  §324 
into  the  cyclic  group  of  e,  gives  a  group  of  quaternions. 

327.  Groups  of  quaternions  whose  coordinates  are  in  an  abstract  field, 
remain  to  be  investigated. 

328.  Theorem.    The  continuous  groups^  of  quaternions  are  as  follows: 

(1)  All  quaternions, 

(2)  All  unit  quaternions. 

(3)  Quaternions  of  the  form  w  •\-  xi  -\-  y% \    Si'^=^Q  =-^-. 

(4)  Quaternions  of  the  form  w-\-y^;  (S'may=:y+  "^ —  1  k). 

(5)  Quaternions  on  the  same  axis,  w  +  xi. 

(6)  Scalars,  rv. 

(7)  The  quaternions  f +i  5(1+  ^/^^l  i)  +  <°  J  (1  — -v^I^l  *)  +  y%, 

t  arbitrary. 

(8)  The  quaternions  e'  +  te'-'^. 

(9)  The  quaternions  1+^3^. 

(10)  The  quaternions  f +  4(1  +  V^^l  i)  +  f  \{\  —  ^/^^i). 

1  ScnEFFBRS  7. 


GENERAL  THEORY  OF  ALGEBRA  75 

Xin.     GENERAL  THEORY  OF  ALGEBRA. 
329.    While  this  memoir  is  particularly  concerned  with  associative  linear 
alyebra,  it  is  nevertheless  necessary,  in  order  to  place  the  subject  in  its  proper 
perspective,  to  give  a  brief  account  of  what  is  here  called,  for  lack  of  a  better 
title,  the  general  theory  of  algebra. 

The  foundations  of  mathematics  consist  of  two  classes  of  things— the 
elements  out  of  which  are  built  the  structures  of  mathematics,  and  the  2mjcesses 
by  which  they  are  built.  The  primary  question  for  the  logician  is:  What  are 
the  primordial  elements  of  mathematics?  He  proceeds  to  reduce  these  to 
so-called  logical  constants  :^  im^iUcation,  relatiun  of  a  term  to  its  class,  notion  of 
such  that,  notion  of  relation,  and  such  further  notions  as  are  involved  in  formal 
nnplication,  Viz. ,  propusitional  function,  class,  denoting,  and  any  or  every  term. 
To  the  mathematician  these  elements  do  not  convey  much  information  as  to 
the  processes  of  mathematics.  The  life  of  mathematics  is  the  derivation  of 
one  thing  from  others,  the  transition  from  data  to  things  that  follow  according 
to  given  processes  of  transition. 

For  example,  consider  the  notions  3,  4,  7.     We  may  say  that  we  have 
here  a  case  of  correspondence,  namely  to  the  two  notions  3,  4  corresponds  the 
notion  7.     But  by  a  different  kind  of  correspondence,  to  3,  4  corresponds  12; 
or  by  other  correspondences  81,  or  -^/H,  and  so  on.     Now  it  is  true  that  in 
each  case  here  mentioned  we  have  a  kind  of  correspondence,  but  these  kinds  of 
correspondence  are  different,  and  herein  lies  the  fact  that  all  corresj.ondcnces 
are  processes.     Equally,  if  we  say  that   we   have   cases  of  relations,— that 
3,  4,  7  stand  in  one  relationship;   3,  4,  12  in  another,  etc.— these  relations  are 
different,  and  the  generic  term  for  all  of  them  is  process.     The  psychological 
fact  that  we  may  associate  ideas   together,  and  call  such  association,  corres- 
pondence, or  relationship,  functionality,  or  like  terms,  should  not  obscure  the 
mathematical  fact,  which  is  equally  psychological,  that  we  may  pass  from  a 
set  of  ideas  to  a  different  idea,  or  set  of  ideas,  —a  mental  phenomenon  which  we 
may  call  inference,  deduction,  implication,  etc.     We  therefore  shall  consider 
that  any  definite  rule  or  method  of  starting  from  a  set  of  ideas  and  arriving 
at   another  idea  or  set  of  ideas  is  a  mathematical  process,   if  any  person 
acquainted  with  the  ideas  entering  the  process  and  who  clearly  understands 
the  process,  would  arrive  at  the  same  goal. 

Thus,  all  persons  would  say  that  3  added  to  4  gives  7,  3  multiplied  by  4 
gives  12,  etc.,  wherein  the  words  add,  multiply,  etc.,  indicate  definite  processes. 
330.    Definition.    A  mathematical  process  is  defined  thus  : 
I.   Let  there  be  a  class  of  entities  \a\. 

II.   Let  there  be  chosen  from  this  class  n— 1  entities,  in  order  a^,a. a„_^. 

in.   Let  these  entities  in   this  order  define  a  method,  F,  of  selecting  "an 
entity,  a„,  from  the  set. 

Then  F{a^,  a,  .  .  .  .  a„_^,  a„)  is  said  to  represent  a  mathematical  process. 


'  B.  RlSSELL  1,  p.  106. 


76  SYNOPSIS  OF  LINEAK  ASSOCIATIVE  AliGBBRA 

The  entities    Oj  . . . .  «n-i    are  called  the  first,  second (n — l)-th 

facients  of  the  process.     The  entity  a„  is  called  the  result.     Occasionally  this 
process  has  been  called  multiplication,  aj  . . . .  a„_ibeing  called  factors. 

331.  The  class  of  entities  \a\  may  be  finite  or  transfinite.  If  transfinite 
they  may  be  capable  of  order,  and  may  be  ordered,  or  they  may  be  chaotic. 
It  is  not  known  whether  there  is  any  class  incapable  of  being  ordered,  or  not. 

The  number  n  may  be  any  number,  finite  or  transfinite,  of  a  Cantor 
ordinal  series  of  numbers. 

332.  Definition.  Let  us  suppose,  in  the  process  F  {a^,  a.;,---  -cin-i,  <^n))  that 
a„  is  known,  but  a^  [l  <  r<  Ji —  l]  is  not  known.  We  may  conceive  that  by 
some  process  F^,  we  can  find  a,,  the  order  of  the  known  terms  being,  let  us 
say, 

K,    «i., (^i^u  «r) 

where  ii,   to-  ■  ■  ••i„-i  are  the  subscripts  1,  2.  .  .  .r  —  1,  r  -\-  \  .  .  .  .n  m  some 
order,  so  that 

^.K,  «i, «i.-,,  «,) 

F„  is  called  d>.  correlative  process,  the  c-correlative  of  i''.     The  process  i^'is  uni- 
form when,  for  all  correlative  processes,  cir  is  determined  uniquely. 

333.  Theorem.  There  are  for  F,  n!  correlative  processes,  including  F. 
We  may  designate  these  by  the  substitutions  of  the  symmetric  group  on  n 
things  ;  so  that  if  we  have 

^(«l,  «2,  «3 ««) 

then  we  also  have 

where  a  is  the  substitution 

3.. 

334.  Theorem.  Evidently  the  cr^-correlative  of  the  cTg-correlative  o^F  is  the 
cTj-correlative  of -F,  where 

(Tg  =  a^  (T2 

We  write,  therefore,  F,-\  ,-1  =  F„-\  =  F^,^,„^-\ . 
The  correlatives  thus  form  a  group  of  order  n  ! . 

335.  Examples. 

(1)  Let  as  be  tax-imijer,  «i  be  hoy,  a^  owner  of  a  dog,  then 
F      (oj  a^a^  :  a  boy  who  owns  a  dog  pays  taxes. 

F(\%)  («i02«3) :  the  possession  of  a  dog  by  the  boy  requires  payment 

of  the  tax. 
F(iz)  («i  ^  f^z)  '•  the  tax  on  a  dog  is  paid  by  the  boy. 
■^(23)  (^1 02  Og)  '•  the  boy  pays  taxes  on  the  dog  he  owns. 
F^■^2;s)  (^1  ^2  (^s)  '•  the  tax  paid  by  the  boy  is  on  a  dog. 
■^{132)  (^i<^2'*3)  '•  the  dog  requires  that  a  tax  be  paid  by  the  boy. 

(2)  Let  aj,  ttj,  03  be  numbers  ;  F{a^^aza^  mean  a^  is  the  a^  power  of  Oj. 


GENEUAL  THEORY  OP  ALGEBnA  77 

Then  ^(,o)  {uicuug)  means  a^  is  the  log  r^^  power  of  the  exponential  of  a,. 
-^'(13)  ("i^u";!)  means  a,  is  the  quotient  of  loga^  by  log  a^. 
i'^ia;),  (ajaoCta)  means  a.,  is  the  a^  root  ofag. 

F,y>;n  (ttj  «.,  U;,)  mcaus  ttj  is  the  log  a,  power  of  the  exponential  of    -  . 

-^(133  ("i  ";; '^'.i)  'neans  Wj  is  log  a^  on  the  Vjase  a.^. 

336.  Theorem.   The  correlatives  of  F  fall  into  sub-groups  corresponding  to 
the  sub-grou])s  of  the  group  (r„, . 

337.  Definition.    It  may  happen  that  in  a  given  process,  F,  we  may  have 
simultaneously  for  all  values  of  cfj. . .  -w,,-! 

^(«i,  «2  ••••«»)  (1) 

i^K,«i- ■••«»)  (2) 

Since   we  must  have  from  (l)  i''^-!  («(_,  a,-,^. . .  .a„)  we  must  identify  F  and 
F^-\,  or  as  we  may  write  it,  F=  F„-i .     Tlie  correlatives  will  break  up  then 

into —  groups  where  ?«   i.s  tlie  order  of  the  substitution  a.     We  call   these 

cases  limitation-types  of  F. 

Examples.     For  i^(aj  n.^  we  have  but  one  case  :  F=.  F^^o,). 
For  F{a^  a^  a^  we  have  five  types  : 

(1)  F^=Fiyi).     This  is  the  familiar  commutativity  of  ordinary  algebra. 
It  follows  that 

-^^(13)  ^^  -'^(23)  -f^(123)  ^  -^(132) 

{2)  Fz=  J?'(,3„  whence  i^j^,  =  F^^^-,,  F^^^  =  F^-^-, 

(3)  F=F^^^,  whence  i^,i,,  =  i^^^oj,,  F^,^,  —  i^^ig.,, 

(4)  i?'=  i^(i2.3,  =  ^(132),  whence  i^dj,  =  F^,^^  =  i^^ag, 

(5)  i^=  i^(i2)     =  i^(,3,  =   7^123,  =  i^|i23)  =  i^(i32) 

For  i''(ai  ttj  c(3  aj  we  have  twenty-nine  types  corresponding  to  the  sub- 
groups of  the  group  G^, : 

(1)  F=  F,,,,  (2)  F=  F,,,,  (3)  F=  F,,,, 

(4)  F=  F,,,,  (5)  F=  F,,,,  (6)  i^=  F,^, 

(7)    i^=i^(j2)^34)  (8)    i?'=  i^„3)(2j)  (9)    -^=i^u),23) 

(10)  i^=Fa23,  =^(i=c)        (11)  F=F,,,,,=  Fa,,,         (12)  J?'^  i?',,3.,  =  i?'.„3, 

(13)    i^=  i^(23,)    =  i^(243)  (14)     J'  =  i^(i234)  =  i^(]3)  (24)  =  -^(14321 

(15)    i*  =  -rii324)  =  -r,]2|  ^34)  =  -^,1423)  (16)    J'    =■  /'(i:j42)  =  i'm)  (23j  =  -^(1243) 

(17)  i^  =  i^jjo)  =  i^;34,  =  i^(i2)  (34)  (18)  F=z  F^^3i  =  F^^i)  =  -^usj  (24) 

(19)  i''=  i''(H|  =  i^(23)  =  i^Ui  (23)  (20)    F=Z  i',12)  (34)  =  i\i3)  (24)  =  -f'iu)  (23) 

(21)  F=  i^(,2)  =  i^(i3,  =  -^(23)  =  -^(123)  =  -^(132) 

(22)  F=  i^|,2,  =  i^(i4,  =  i^(24)  —  i^,i34)  =  i^(i42) 

(23)  F=  i^(i3)  =  i^i4,  =  J^(34,  =   7^,134,  =  i^i43) 

(24)  F=  i^(23)  =  F^2l)   =  i^i3t)  =   -^(23i)  =   -^1243) 

(25)  J^  =  F^^^|  =  F^o^^  =  F^i^-^  (24)  =  -''(1234)  ^^  -^(12)  oo  =  -^(U32)  ^^  -^lU)  (23) 

(26)  7^^:=  i'^ij)  =  i^34)  =  -r\i2)(34)  ^=  -^(1324)  ^^  -^(13)  (24)  =  -^1423)   ^  -^(14)  (23) 

(27)  F^  F^^^^  =  i'iga)  =  F^^^  ^3^■^  =  -rjiigs)  =  -P(W)  (gs)  =  -f^i824)  =  -^(13)  an) 


78  SYNOPSIS  OF  LINEAR  ASSOCIATIVE  ALGEBRA 

(28)  i^=  Fq2)  (34)  =  -^(13)  (24)  ^  -^(23)  (14)  =  -'^ISS)  =  -^'^(132)  =  -^(124)  =  -^(142) 

-—  -^(134)  ^  -^(143)  — -  -''(2ai)  ^  -''(243) 

(29)  i^=all 

338.  Theorem.  It  is  evident  that  every  group  defines  a  limitation  type  for 
an  operation  F  of  some  degree. 

339.  Definitions.  Suppose  that  in  a  process  all  the  elements  but  two  are 
fixed,  and  that  these  two  vary  subject  to  the  process.  Then  the  ranges  of 
values  of  these  two  are  said  to  form  an  involution  of  order  one.  If  all  but 
three  elements  are  fixed,  the  ranges  of  these  three  form  an  involution  of  order 
two.     Similar  definitions  may  be  given  for  involutions  of  higher  order. 

An  involution  of  order  r  is  often  called  an  implicit  function  of  r  +  1 
variables.  The  symbol  consisting  of  the  process  symbol  and  the  constant 
elements  is  called  an  operator. 

If  in  any  involution  of  order  one  the  two  elements  become  identical  so 
that  they  have  the  same  range,  for  any  given  set  of  constant  elements,  then 
this  set  of  constant  elements  constitutes  a  multiplex  modulus  for  the  process. 

For  example,  in  multiplication  F.{ah  =  h)  when  a  is  1.  A  similar  defini- 
tion holds  for  higher  involutions. 

If  in  any  involution  of  order  r,  the  constant  terms  determine  an  involu- 
tion whose  terms  may  be  ani/  elements  of  the  set,  then  the  constant  terms 
constitute  a  zero  for  the  process.  For  example,  if  F.  (Oa  =  0),  for  all  a,  0  is  a 
zero  for  multiplication.     An  infinity  is,  under  this  definition,  also  a  zero. 

We  have  seen  that  there  co-exist  with  any  process  i^ certain  other  correla- 
tive processes  on  the  same  elements.  These  give  us  a  set  of  co-existences 
called  fundamental  identities;  but  we  may  have  co-existent  processes  which 
are  not  correlatives.     In  the  most  general  case  let  us  suppose  that  we  have 

F'  .Uiittiz «i„,  F"  .a^^a^ az^, -?""'"" -flr+i,!  «,-i, a ^''•-i."r_i 

and  that  when  these  processes  exist,  then  we  have  F^'"'  .  a^.^  a,., a,.„, . 

We  say  that  i^^'''  is  the  implication  of  the  r  —  1   processes  preceding.     We 
enter  here  upon  the  study  of  logic  proper.     For  example,  if  the  processes  are 

F' .  ah  F"  .  ho  F'"  .  ac 

we  have  the  ordinary  syllogism. 

We  may  symbolize  this  definition  by  the  statement 

*  .  i^,'„  F'l_ FtZ^  F^C 

and  we  see  then  that  the  form  is  again  that  of  a  process  *. 

We  can  not  enter  on  the  discussion  of  these  cases  beyond  the  single  type 
we  need,  called  the  associative  law. 

Let  F  be  such  that  for  every  a,  h,  c,  we  have 

F.ahd  F  .dee  F  .  beg         then  F .  age 

then  F  is  called  associative.     The  law  is  usually  written  ah  .c  =  a.  be 

Processes  subject  to  this  law  are  the  basis  o(  associative  algebras.' 

'Cf.  SonaoEDEB  1;  Rcssbll  1,  2;  Hathaway  1. 


=  0 


PART  II.    PARTICULAR  ALGEBRAS. 
XrV.     COMPLEX   NUMBERS. 

340.  Definitions.    The  algebra  of  ordinary  complex  nuinbers  possesses  two 
qualitative  units,  eQ=  1,  ami  e,,  such  that 

The  field  of  coordinates  is  the  field  of  positive  and  negative  numbers.     The 
field  naturally  admits  of  addition  of  the  units  or  marks. 

341.  Theorem.    Tlie  characteristic  equation  of  the  algebra,  as  well  as  the 
general  equation,  is 

'^^  —  2x^  +  x"  +  if—0 
or 

—  y   icco— ^ 

Hence  for  any  two  numbers 

^(T  +  (T^  —  2X(T  —  2/^  +  2xx'  -f  1>JlJ  =  0 
or 

^<T  —  xa  —  xX  +  xx'  -\-  yij  -=.0 

The  characteristic  equation  is  irreducible  in  the  field  of  coordinates  but 
in  the  algebra  may  be  written 

(^  —  a-Co  —  ye^)  (^  —  xCq  +  ye^)  =  0 

The  numbers  ^  =:  arco  +  ye^  and  iT^  =  ^=  a-Co  —  ye,  are  called  conjugates. 
Hence  ^2  _  2x  ^  +  x^  +  /  =  0  has   the  two  solutions  ^,  K^,  or  (^  —  ^) 
(^  —  KQ  are  its  factors. 

342.  Theorem.    If  several  algebras  of  this  kind  are  added  (in  the  sense 
defined  by  Scheffers)  we  arrive  at  a  Weierstrass  commutative  algebra. 

343.  Theorem.    If   the    coordinates  are   arithmetical   numbers    we    must 
write  this  algebra  as  a  cyclic  algebra  of  four  qualitative  units 

Cq      Ci      e.,      eg 
where 

e\  =  ^2  e?  =  63  e\  =  e^ 

In  this  case  the  units  Cq  and  e,  are  not  independent  in  the  field,  and  com- 
bine, by  addition,  to  give  zero,^  and  the  algebra  is  of  two  dimensions. 


'  Study  S,  and  references  there  given;  Beuax  3;  Bellavitis  1-16 ;■  Bibliography  of  Quaternions. 

79 


80  SYNOPSIS  OF  LINEAR  ASSOCIATIVE  ALGEBRA 

XV.     QUATERNIONS. 

344.  Definition.  Quaternions  is  an  algebra  whose  coordinate  field  is  the 
field  of  positive  and  negative  numbers,  and  whose  multiplication  table  is 
(«o=l) 


^0 

ei 

«% 

^3 

Co 

'-0 

ei 

^2 

% 

^1 

ei 

—Co 

^3 

— ^2 

^2 

^2 

—es 

— ^0 

ei 

eg 

«3 

ez 

— ei 

—  <'o 

345.  Theorem.    The  characteristic  equation  is 

^  —  2xo^  +  a-o  +  XI  +  icl  +  a|  =  0 
The  characteristic  function  of  ^  is 

f— 2xo^  +  ^^  +  :^  +  aH-a:i 

If  we  define  the  conjugate  of  ^  by 

then  the  characteristic  function  factors  into  (^ —  ^)  (^  —  Z"^). 

346.  Theorem.    The  first  derived  characteristic  function  of  ^  and  a  is 

(.  -Q{a-  K^')  +  (cT-  ^0  (^  -  ^0 
This  vanishes  for  ^ v  y/ 

'^''  ^  =  /f^         a  =  K^' 

347.  Definition.    The  scalar  of  ^  is  ^^  =  .to  =  ^  (^  +  ^).     The  tensor  of  ^  is 
given  by  (7'0^  =  Cr=  ??• 

348.  Theorem.    We  have 

^T  +  T^  —  2^S^'—  2rS'(  +  2FF^7^'  +  2  FFtF^  +  2  ^^If'  =  0 
if  ^  =  ^,  and  T  =  ^';  or  if  ^  =  ^,  T  =  ^  ;  also  S^^'  =  S^^'. 

r 

349.  Definitions.    The  versor  of  ^  is  U^  =  -^     The  vector  of  ^  is 

350.  Theorems.    ^  =  >S^  +  F^^         ^  —  S^  —  F^ 

( T'O'-  =  ( ?t?  =  (-^O^  -  ( VO'  =  {S^f  +  {  TV'(f 

S. '(0  =  801:  S.  K^  =  KS.(  =  S^ 

If  S(=0  =  Sa  a  =  Va     ^  =  V( 

and  F^.  Fa  =  — Fct7^+  2^.FffF^ 


QUATERNIONS 


81 


Also  KV^  =  -  F^  =  VJit; 

'(  .  Va^-'  =  —  Va  +  2jS(;.Va.^-'+2  ^-'  JS^Va 

z=  —  Va  +  2S(.V^^  +  2V.^-'  ,Sr(V>y 
If  a,  /?,  y  are  vectors, 

F  .  a  V(3y  =  ySafS  —  pSay  V .  a^y  =  aSi^y  —  (iSya  +  ySa^ 

If  ^  is  a  vector, 
hSaPy  =  aSpyh  +  pSyah  +  ySa^h  =  Va(3Syh  +  VPySah  +  VyaS^h 

V  Fa/?  VyS  =  hSa^y  —  ySa(3b  =  aS/Syb  -  ^S<x.yh 
If  a,  h,  c,  d,  e  are  quaternions,  let  us  use  the  notation 


Also 


Then 


B  .ah  =  i  {ab  —  ha)  B<.  ah  =  bSa  —  aSb 

B  .ahc  =  S  .  aBhc  —  V{aBhc  +  hBea  +  cBah) 

B  .  ab  =  B  .  ah  =  V  .  VaVh  =  —  B  .  ha  B  .hh  —  0 

S  .  aBbc  =  S  .  VaVhVc=  —  ShBac  =  etc. 
B  .  ahc  =  —  B  .  hac  =  etc. 
S  .  aBbcd  =  —  S  .  hBacd  =  etc. 
e;iS' .  aBbcd  =.  aS  .  eBbcd  —  hS .  eBcda  +  C/S .  eBdab  —  dS .  eBahc 

=^  —  Sde  .  Babe  +  Sae  .  Bbcd  —  She  .  Bcda  -\-  See  .  Bdah 

c         d         e 
Sac     Sad     Sae 
She     Sbd     Sbe 


B  .  ahBcde  = 


B  {Babe  Bdef  Bghi)  = 


B'{BabcBdef)  =      BefSaBbcd  +  BfdSaBbce  +  Bde  SaBhcf 
B  {BabcBdef)  =—  BefSaBbcd  —  B'fdSaBhce  —  Bde  SaBhcf 

ahc 
SaBdef      SbBdtf       ScBdcf 
SaBgJd      ShBgJd       ScBgJti 

I  Saa',  Sbh\  Sec'  I  ^  —  SBahc  Ba'b'd 

I  Saa',  ShV  I  =       SB'ah  B'a'h'—  SBab  Ba'U 
I /Saa',  SbU,  Sec',  Sdd'\  =  —  SaBhcd  Sa'  Bb'c'd' 

The  solution  of  the  equation  aip  +^a2  =  c  is 

pz=z.{a\-\-  2aj  Sa^^  +  a„  Ui)~^  (cTj  c  +  ca.^ 

351.  Definition.  If  the  coordinate  field  contain  the  imaginary  V — 1,  we 
may  have  for  certain  quaternions  the  equation  (f  =  0,  whence  q  =  yd,  ()~=.  0, 
and  y  is  any  scalar.  In  this  case  there  is  an  infinity  of  solutions  of  the  equa- 
tion in  the  algebra. 


82 


SYNOPSIS  OF  LINEAR  ASSOCIATIVE  ALGEBRA 


Also  if  (f  —  20^0 g-  +  jcq  =  0,  then  q-=.Xf^-\-  yQ. 

The  nilpotent  Q  is  always  of  the  form  a  +  */ —  1  /?,  where 

a^  =  /^-  S.aP  =  0 


Since  V —  1  will  not  combine  with  positive  or  negative  numbers  by 
addition,  we  may  say  that  this  algebra  is  in  reality  the  product^  of  real 
quaternions  and  the  algebra  of  complex  numbers,  giving  the  multiplication 
table 


€o 

ei 

^2 

e^ 

^4    ej 

«6 

(°7 

eo 

^0 

ei 

^3 

es 

^4    ^5 

^6 

Cj 

«! 

ei 

— eo 

«3 

—  ^2 

es  — e* 

^7 

—  ^6 

«2 

e.. 

—  ^3 

— eo 

f"! 

es  — e^ 

— e* 

^5 

ez 

^3 

^2 

— ei 

—  eo 

^7  +^6 

—  ^5 

—  64 

«4 

ei 

fs 

^6 

^7 

Cq  ^I 

^^3 

—  ^3 

^5 

^5 

—  ^4 

^7 

— f-e 

— ei  +^0 

— es 

+e. 

<?6 

•"e 

—  ^7 

— e* 

<'5 

—  ^2  +^3 

+  ^0 

— ei 

e-i 

er. 

+  ^6 

— ^5 

— e^ 

^3  — ^2 

+  ei 

+  ^0 

with  equations  of  condition 

eu  +  e*  =  0  cj  -f  cs  =  0  e^  +  eg  =  0 

This  algebra  Hamilton  called  Biquaternions. 


^3  +  f  7  =  0 


352.  Definition.  The  algebra  ?.jio,  Xjoq,  /Lojo,  ^220  fi^so  is  a  form  to  which 
real  quaternions  may  be  reduced  by  an  imaginary  transformation.^  By  a 
rational  transformation  this  becomes 


'^uo  +  ^'. 


220 


?. 


110' 


^220 


^^210  T  '^12( 


120 


Aoin  ~~"  Ai 


120 


References  to  the  literature  of  quaternions  would  be  too  numerous  to  give 
in  full.  They  may  be  found  in  the  Bibliography  of  Quaternions.  In  particu- 
lar may  be  mentioned  the  works  of  Hamilton,  Tait,  and  Joly. 

I  That  is,  any  unit  may  be  represented  by  a  double  symbol  of  two  independent  entitles,  the  two  sets 
of  symbols  combining  independently. 
'  B.  Feirce  3. 


ALTERNATE}  ALGEBRAS  83 

XVI.     ALTERNATE  ALGEBRAS. 

1.  ALGEBRAS  OF  DEGREE  TWO,  WITH  NO  MODULUS. 

353.  Definition.    An  ulternate  algebra  is  one  in  which  the  defining  units 
are  subject  to  tlie  law 

The  product  e^  Cj  =  e\  is  variously  defined.    In  the  simplest  cases  c?  is  taken 
equal  to  zero. 

354.  Theorem.    When  Ci e^  +  e^ei  =  0,     i,  y  =  1 . . . . r,  we  have 


e?  =  0  1=1.. 


r 


^2=0,  ^(T  +  <T^  =  0  all  values  of  ^,  a 

^ a T  =  0  all  values  of  ^,  a,  t 

For 

and' 

^(T  .  T  = ^  .  Tff  =  TT  .  ifff  =  —  T  .  ^(T  =  0 

355.  Theorem,  We  may  therefore  select  a  certain  set  of  r —  m —  h  units, 
ej  .  . .  .  Cr-iH  A>  whose  products  CiCj^i,  j=  1  .  .  .  -r  —  m — h)  are  such  that  at 
least  one  for  each  subscript  does  not  vanish ;  we  may  then  choose  for  the  next 
m  units  the  ?n  independent  non-vanishing  products  of  the  first  r — m  —  h 
units;  finally,  the  last  h  units  may  be  any  numbers  independent  of  each  other 
and  the  first  r  —  h  units.     W^e  must  have  ^ 


„,  <  2(r-h)+l-VS{r-h)+l 
2 
or 

(r  —  m  —  hy  —  (r  -\-  m  —  A)  =  0 

2.     GRASSMANN'S    SYSTEM. 

356.  Definition.  The  next  type  of  alternate  numbers  is  that  of  Grassmann's 
Aiisdt'linungslehrc.  In  this  case  there  are  m  units  wbich  may  be  called  funda- 
mental generators  of  the  algebra,  Cj  .  . .  .  e,„.  For  them,  but  not  necessarily  for 
their  products,  the  law  e^Cj -\- ej  6^-:=  0  {i,  j ■=■  I. .  . -m)  holds.  They  are 
associative,  and  consequently  the  product  of  ttj  +  1  numbers  vanishes.  There 
are  r  =  2'" —  1  products  or  units,  e,,  e^e^,  CjCje^,  etc. 

This  algebra  uses  certain  bilinear  expressions  called  products,  which  do 
not  follow  the  associative  law,  and  also  certain  regressive  products,  which  do 
not  follow  this  law,  and  which  are  multilinear  expressions  in  the  coordinates 
of  the  factors.^ 

1  ScHEFFERS  3.     Cf.  Caucht  1,  3,  S  ;   Scott  1,  2,  3.  <ScHErFBBS  3. 

•References  are  too  numerons  to  be  given  here.  In  particular  see  Sibliography  of  Qualemiont; 
Grassmann's  works;  Schleoel's  papers;  IItdh  1,  3,  3,  4,  5,  6,  7,  8,  9 ;  Beman  1;  Whitehead  1.  Cf. 
WiLSON-GiBBS  1 ;  Jahnke  1.     Works  on  Vector  AnalysU  are  related  to  this  subject  and  the  next. 


84  SYNOPSIS  OF  LINEAR  ASSOCIATIVE  ALGEBRA 

3.     CLIFFORD   ALGEBRAS. 

357.  Definition.  A  type  of  alternates  of  much  use,  and  which  enables  all 
the  so-callecl  products  of  the  preceding  class  to  be  expressed  easily,  is  that 
which  may  be  called  Clifford's  Algebras.  Any  such  algebra  is  defined  by  m 
generators  e^. . .  .e,„  with  the  defining  equations 

ef=  — 1  Ci  Cj -I- e,  Ci  =  0         »,/=! m;  i:^j 

Ci .  Bj  e^  =  Ci  Cj .  e^t  i,  y.  A;  =  1 . .  . .  m 

The  order  Ms  r  =  2'". 

358.  Theorem.  If  m  =  2ni',  m'  an  integer,  the  Clifford  algebra  of  order 
2'"  =  4"' is  the  product  of  m'  quaternion  algebras.  It'  m  =  2m'  -{-  1,  m'  an 
integer,  the  Clifford  algebra  of  order  2™=  q^"'"*"^  is  the  product  of  m'  quater- 
nion algebras  and  the  algebra^ 

2  __     o  ^^  __^ 

359.  Definition.    Since  any  product  such  as 

^i,  ^i;  •  •  •  •  ^ii  •  •  •  •  ^i,  •  •  •  • 

may  be  reduced  by  successive  transpositions  to  a  product  of  order  two  lower 
for  every  such  pair  as  e^, . .  •  -e^j,  it  follows  that  in  the  product  of  n  numbers 

^1-  •  •  -^nt  where  ^4=2  x^e,,  we  may  write 

S 1  SZ  •  •  •  •  S  n  =   T  n  •  S 1  •  •  •  •  b  II  4"     •«  -  3  •  S I  •  •  •  •  Sn  +  •  -  -  • 

<  J  (n  +  1) 

8  =  0 

The  expression  F„_2j,  is  the  sum  of  all  expressions  in  the  product  ^i....^„ 
which  reduce  to  terms  of  order  n  —  2s.  Evidently  when  n  is  even,  the  lowest 
sum  is  a  scalar,  Fqj  when  n  is  odd,  the  lowest  sum^  is  Fj. 

360.  Theorem.  To  reduce  to  a  canonical  (simplified)  form  any  homo- 
geneous function  of  A'' of  the  m  units,  consisting  of  terms  which  are  each  the 
product  of  a  scalar  into  n  of  the  units  (of  order  n,  therefore),  we  proceed  thus : 

Let  q  be  the  given  function.  Then  by  transposing  the  units,  we  may 
reduce  q  to  the  form 

q=  —  q'  ii  +  q" 

where  i\  is  any  given  unit,  and  q'  (of  order  n  —  1)  is  independent  of  q"  (of 
order  n)  and  of  ij.     We  find  easily 

q'  =  F„  _  J  qii  q"  ij  =  F„  +  i  qi^ 

Therefore 

Fi .  <?2'  =  Fj  <?  F„  _i  qii  =  wj  =  O  (ii) 

The  linear  vector  function  <!>  is  self-transverse,  has  therefore  real,  mutually 
orthogonal  axes.     These  are  the  units  to  be  employed  to  reduce  to  the  canon- 

'For  this  class  see  Sihliography  of  Qualernions;  in  particular  Clifford's  works;  Beez  1 ;  Lipsohitz  1; 
Jolt  6,  12,  25 ;  Cati-et  6,  7. 

'TabbrI.  'Jolt  6. 


ALTERNATE  ALGEBRAS  g5 

ical  form.     For  example,  if  q  is  of  order  2,  and  the  function  is  the  general 
quadratic  for  N  units,  there  are  i  N{N —  1)  binary  products.     Then 

—  m^=  F]  qi  Vq^i  =  —  ^i  ™i  +  V^q'  a  =:  —  V^q  Vj  qi^  =  —  4>ti 

If  ij  is  an  axis  of  this  equation  so  is  Wj.     Hence  the  quadratic  takes  the  form 

q  =z  aj3  ^1  *2  +  «(n  ^3  *4  4"  •  •  •  •  +  (tzp  - 1, 2p  hp  - 1  hp 
where  2>  =  ^  N  or  i  {N —  1)  as  iV^is  even  or  odd.' 

361.  Definition.    Let  K  change  the  sign  of  every  unit  and   reverse  every 
product.     Tiiea  ii' q  is  liomogoneous,  of  order  p, 

K.q,  =  {-yr><v  +  ^^q, 
Hence  K.  qp=  ±  q^  as  /)  =  0  or  3  (mod  4)  or  =  1  or  2  (mod  4). 
Let  /reverse  the  order  of  products,  but  not  change  signs,  thus 
I.gp  =  qp  if  p  =  0,  or  1  (mod  4) 
/.  qpZ=  —  q^^  if  J)  =  2,  or  3  (mod  4) 

Let  J  chang'e  the  signs  of  units  but  not  reverse  terms. 

362.  Theorem.    K.  pq  =  Kq  .  Kp         I.  pq  =  Iq.Ip        I  =JK=  KJ 

J  =  Kl  =  IK  K  =  1J=  J I  P  =  J'  =  K-=1 

363.  Theorem.    Let  j>  be  of  order  2,  q  of  order  3  ;  then 

Pz  ?3  =  1^1  •  2h  ?3  +  Fj  .  p2  ^3+^5.  pz  qs 

Hence,  taking  conjugates, 

—  QaPz  =  —  Vi-  2h  ?3  +  ^3  •  p2  qz—V^.  Pi  53 
and 

^1  •  i'g  S'a  +  "^6  •  i'3  g-s  =  ^  {pi  ?3  +  qzP^  ^a  •  ih  qa  =  \{  ih  qs  —  qa  Pi) 

This  process  may  be  applied  to  any  case. 

364.  Theorem.    Let 

q  =  q'  +  q"  K.q  =  q'—q" 

q.Kq=q'"'-q"^~-{^q"-q"<^) 
Kq.q^f^-i'-'^islci'-ci'cl) 
Hence 

q  .  Kq=  Kq  .q  if  5'  q"  —  q"  q' =  0 

Let  the  parts  of  q  be  (according  as  their  order  =0,  1,  2,  3  mod  4) 

q  =  ?(0)  +  ?(!)  +  qm  +  ?(3) 

Then 

q"  q"  =  q^o)  qa)  +  q^)  ?(2)  +  q^o^  qi?)  +  q,z)  qa) 

and  the  condition  above  reduces  to 

9(")  qm      9(1)  9(0)  =  q(2)  9(3)      9(3)  9(2)  9(0)  9(2)      9(3)  9(0)  =  9(i)  9(3»  —  9(3,  q^) 

or 

^(3)  (9(0)  q^i)  —  9(3)  9^3))  =  0  1^(0)  (9(0)  9(2)  —  9(1)  93))  =  0 

'Jolt  6.     This  reference  applies  to  the  following  sections. 


86  SYNOPSIS  OF  LINBAK  ASSOCIATIVE  ALGEBRA 

When  this  is  satisfied 

qKq  =  F,o,  (9^0)  —  5a)  —  ?(1)  +  ?J,)  +  2 1^,3)  iq,o)  9(3)  —  9a)  9,3 ) 
This  is  a  scalar  if  T^Q)  —  ^o  ^^^^  F^^^  =  0. 

365.  Theorem,    q-  Iq  =  Iq  -q    if 

^(0)  (9(0)  qrz)  —  9(3)  qa^  =0  =  F^i>  (g^„)  (^^3,  —  g-,.,  ^d,) 

366.  Theorem.    Let   P=  qpq~^,  where  q  is  any    number,   possibly  non- 
homogeneous.     Then  P=.V^i^  .  P  if  qKq  and  qlq  are  scalars. 

But  F(i)  may  not  =Fi.     For  example,  let 

q  z=:       cos  ?<  .  t'l  ?2  +  sin  ?{  .  ig  1*4  ig  ig 
q~^  =  —  cos  u  .  ii  u  +  sin  u  .  i^  i^  ij  if, 
q"  =:  —  cos"  u  +  sin-  zt  +  2  sin  u  cos  u  I'l  i^  ^3  i^  1*5  ig 

p  =  ij,  i.,  is,  ii,  if,,  is,  then 

P  =  q\         q~u         —  2^*3         —  5'\         —  5-%         —  <i\ 
which  are  of  the  form  F^d  z=  Fj  +  Fg. 

367.  Theorem.    An  operator  5 O^"' can  be  found  which  will  convert  the 
orthogonal  set  i^,  i^.  .  .  ■  i„  into  any  other  orthogonal  set  ji,jo  •  •  ■  ■  jn\  namely, 


where 

s,  t,  u  .  . . .  =  1,  2  .  .  .  .  n  s:^t:^u:^  .  .  .  . 

If  qpq~^  ^=^Ppi>~^  foi'  all  values  of  p,  a  vector,  then  j  is  a  scalar  multiple 
of  p.     q  may  be  written 

9l2  9»» 931-1.21 

where  2l  =  n  or  7i  —  1  as  m  is  even  or  odd,  and 

grj2  ^  cos  ^  Mjo  +  h  h  sin  ^  ?<i2,  etc. 


BIQUATERNIONS  OR  OCTONIONS  87 

XVn.     BIQUATERNIONS  OR  OCTONIONS. 

368.  Definition.  Besides  Hamilton's  biquaternions,  two  algebras  Lave 
received  this  name.  One  is  the  product  of  real  quateriiiotis  and  the  algebra 
6i :  e5  =  eo  =  e^,  pq  Cj  =  ejeo  =  Cj;  the  other  is  the  product  of  real  quater- 
nions and  the  algebra '  63 :  e^=  Cg,     eg  Sj  =  e,  «„  =  ''1,     ej  =  0. 

369.  Definitions.  Let  fl"  =  0  ;  let  q,  r  be  real  quaternions  ;  11  is  commu- 
tative with  all  numbers;  q  =  G>-{.x;  r  =  a+i/.  Then  the  octonion  Q  is 
given  by 

Q  =  q  +  nr 

We  call  q  the  axial  of  Q,  Clr  the  converter  of  Q.     The  axis-direction  of  Q  is 
UVq.     The  perpendicular  of  Q  is  m  =  F .  ctcj-^     The  rotor  of  Q  '\s  Vq;    the 
lator  is  Fr;  the  »»o/or,  Vq  +  H  Fr.     The  ordinary  scalar  is  iS'*/;  the  scalar-con- 
verter is  fl/Sr;  the  convert  is  *S'r. 
We  write 

J/i  Q  =  Vq         M,Q  =  £iVr        MQ  =  M,  Q  +  M„  Q 
m.  Q  =  Vr  S,Q  =  >Sq         S,  Q  =  flAV  SQ  =  S,  Q -\-  S..  Q 

s.  Q  =  Sr         M.Q  =  M,Q  +  £imQ  S .  Q  =  Sy  Q -^  D.Iq 

Let  y,  r,  Q  be  the  conjugates  of  q,  r,  Q,  also  designated  by  Kq^  Kr,  KQ. 
We  define 

KQ  =  Kq-\-  D.Kr,  or   Q  =  q-\-D.? 

The  tensors  of  q  and  r  are  y'g-,  Tr ;  the  versors,  Uq,  TJr : 

The  avgmenter  of  $  is  7g=  r*?  (1  -f  nAS'r5-')=  7',  Q.T.,Q=T,Q  {l+HtQ). 

The  ^e«6o?-  of  g  is  T;  §. 

The  additor  of  §  is  7;  g  =  1  +  nxS'ry-i. 

The  pitch  of  Q  \s  tQ  —  S  .  rq-\  T„Q=1  -\-  flfQ. 

The  twister  of  Q  is  UQ  =  Uq  {1  -^  D.  Vrq-^}  =  tr,  g  .  UQ. 

The  verso/-  of  Q  is  ZJj  g  =  Uq. 

The  translator  of  $  is  t^o  §  =  1  +  11  Fry-'. 

Hence 

g=7;g.7',g.  f/-, g.  tr,g 

370.  Theorem.  Octonions  may  be  combined  under  all  the  laws  of  quater- 
nions, regard  being  given  to  the  character  of  il. 

371.  Theorem.    If  Q,  R  be  given  octonions 

Q-^R  =  X  QE=Y 

and  if  s  is  any  lator;   then  if 

Q'=  Q  -h  £lMeMQ  R  =  R+  (^ME}fR 

then 

Q'  +  R'z=  X'  Q<  R'  =  Y' 


'Clifford  1,  -J  ;    M'Aulat  2,  wbich  applies  to  sections  foUowing;  Combebiac  1,  2;  Stcdt  4    5 


88  SYNOPSIS  OF  LINEAR  ASSOCIATIVE  ALGEBRA 

the  application  of  ^,  to  all  octonions  gives  an  isomorphism  of  the  group  of  all 
octoaious  with  itself. 

372.  Theorem.    If  Q  =  ^, .  Q',  or  q  +  D.r=^^  (q'  +  Hr'),  then 

q'  =  q  r'  =  r  —  i/(p  Mq) 

373.  Definition.    The  axial 

q(i  =  q  +  D.  {Mqy^  MMqMr  =  a:  +  o  +  0(0"^  Mi^a 
is  called  the  special  axial  of  Q,  and 

r^  =  Sr+{\  +  CiMMr  ( Mq )-')  Mq  S Mr  ( Mq)-^ 

-  y  +  Tl  +  £iMM  ^  ")    ioS  — 
is  called  the  special  convertor-axial  of  §. 

374.  Theorem.    We  have 

Q  =  q  +  nr  =  qg  +  Hz-g 

"""^'^'^  q,=  \l+M.{Mr[Mqr'){)\q  =^M...^^q 

r^  =r  -;  1  +  1/  .  (i//-  [Mq]-')  ()\lr—3£.  {Mr  [Mq]-')  Mq) 
=  <pM  .0.-1  (r  +  J/ .  cjJfocj-^)  =  <?).v  ,.-1  (aS/-  —  a-'  *S'a)(7) 

That  is 

$  =  ^M.  ..-1  ?',  where  Q' =  q  +  D.  {Sr  —  cj-'  &>a) 

or 

Q'=  q+n  [Sr  —  {Mq)-'  SMq  i/r] 

375.  Theorem.  Any  octonion  mny  be  considered  to  he  the  quotient  of  two 
motors.  That  is,  if  Q  be  an  octonion  it  may  be  written  Q  =  BA"'  or  QA  =  B, 
where  A  and  B  are  a  pair  of  motors. 

376.  Theorem.     Q'  =  q-'—^q-'r q^\  when  q:^0. 

377.  Definition.    The  angle  of  g-  is  fJie  angle  of  Q. 

378.  Theorem.  Q  ()  Q~'  produces  from  the  operand  a  new  operand  which 
has  been  produced  from  the  first  by  rotating  it  as  a  rigid  body  about  the  axis 
of  Q  through  twice  the  angle  of  Q,  and  translated  through  twice  the  transla- 
tion of  Q. 

379.  Theorem.    If  A  and  B  are  motors 

A  =  a^  +  na2  =  (l  +  Clp)  a, 

B=(3,  +  a(3,  =  {\+  ap')[i,  =  ]  1  4  n(m  +/)}/? 


and 
then 


AB  =  ai^  +  a{p+p'  —  w)ai(S 
M .  AB  =  Ma,p  +  il  \{p  +  p')  Ma,p  —  w  Sa,^\ 
M,  AB  =  Ma,lJ       m.AB  =  {p  +  p')  Ma,  l3—wSa,/3 
tM .  AB  =p-{-p'  —  m  M-'  UiiSSui  (3  =pi-p'  +  dcot  6 
Hence  axis  M.  AB  is  w,  pitch  =p  \-  p'  -^  d  cot  d 


BIQUATERNIONS  OR  0CT0NI0N3  89 

If  A  and  B  are  parallel,  we  deteniiiiie  M .  Ali  by 

M.AB  =  —  D.ma^^ 
Again  S.AB  =  S.a,(i  +  £l\{  p  +  p')  .S'a,  /?  —  -^  Ma,  [i  \ 

S,  AB  =  S<x,  (3 

s.AB=(j>  +  //)  ,Va,  (i—w  Mai  /? 
tS.AB=p  +  j,'—m  Mai  t^  'S~^  "1  f^  =  l'  +  p'  —  d  tan  0 
Mi.AB  +  S,.AB  =  ai(i 

m  AB  +  sAB={pJr  p'  —  m)  a,  P 

tAB  =  p  +  p'        u.AB  =  —  m 
T,.AB=  T{u,  (3)      U,AB=  U{a,  /?) 
For  the  sum  we  have 

A-\.B  =  \l+D.  (//'  +  u,')\  (a,  +  li) 

where  p"  +  w' =  (p  a,  j- p' (3  +  w  13)  (a,  +  (3)-' 

or 

p"=S{ l> a,  +p'l3)  (a,  +  /3) -'  -  u,  i/ai  13  .  (a^  +  /3)-2 
m'  =  w  Si3  (aj  +  f3r'  +  {p~  p')  Ma,  /3  .  («!  +  /i)-- 

380.  Theorem.  If  .4,  B,  C  be  three  motors,  and  if  cZ  and  0  are  defined  as 
in  §.379  for  A,  B,  and  likewise  e,  ^  are  corresponding  quantities  for  M .  AB 
and  G,  then 

^  SABC=  (A  +  tB  +  tC+d  cot  0  —  e  tan  4) 

Hence  if  we  have  three  motors  1,  2,  3,  and  if  the  distances  and  angles 
are:  for  23  :c/j,  gj;  for  31  :  (7o,  0^;  for  12  :  cZg,  0;;,  and  fjr  1  and  (Zi :  e^ ,  ^,;  2  and 
rfo :  e.,,  ^.r,  3  and  d-^'.e^,  <p3,  tlien 

d,  cot  6i  —  Ci  tan  ^j  =  d.^  cot  0,  —  e.^  tan  ^3  =  tZj  cot  63  —  e^  tan  ^3 

381.  Theorem. 

T,{QR  ....)=  T,Q.T,R....  ( (QR  ....)  =  tQ  +  tU +... . 

S\Q-M\Q=T\Q  tSQ.S\Q~tMQ.M\Q  =  tQ.  T\Q 

A  .nrr,       tQT\Q  —  tSQ.S\Q 

and  tMQ  =  -^gZT^Ig-^^ 

382.  Theorem. 

i  Tir    A  T>ri      J  <    I    .  r.  I    ,  ^  <^  cot  Q  —  e  tan  A 

cot-0tan-^ +cot-0  +  tan-(^ 

tM.  [MAB)  C=  fA  +  /5  +  tV  +  (Z  cot  0  +  e  cot  (^ 

383.  Theorem.    If  £■  is  coaxial  with  A,  B,  C,  then 

ES .  ABC  =AS.BCE-\-  BS.  CAE  +  CS .  ABE 

=  MBC.  SAE+  MCA  .  SBE  +  MAB .  SCE 


90  SYNOPSIS  OF  LINEAJl  ASSOCIATIVE  AT^GEBRA 

384.  Theorems. 

S,{Q  +  E)  =  S,Q+S,R  s{Q  +  R)  =  .sQ  +  sR 

SQ  =  sQ.Q  +  £lsQ 

If  iljl k,  then 

A  =  —  iSiA  —  jSJA  —  kSkA 
or 

A  =  —  isD.iA  —  JsD.jA  —  ks£ncA  —  £iisiA  —  0.jsJA  —  D.kskA 

If 

A  =:  xi -\-  yj  -\-  zk  ■{■  lD.i  +  mD.j  +  7iQ.k 
and 

cs       .  d    ,    .  d     ,,9,„.9.„.9        ^,9 
^=^9T+-^9^+^9^  +  "^9^  +  "'^9^  +  ^^^97 
then 

{s.dAb)  =  —  d{) 
S'  is  independent  of  i,j,  k. 

If  ^  is  a  lator,  s  .  A-  =^0. 
If  A  is  not  a  lator 

s.A"  =  2tA3IlA  =  —  2tAT]A  s{TQY=  2tQ  .  T\Q 

385.  Definitions.  The  motors  Ai,  A.,....A,i  are  independent  when  no 
relation  exists  of  the  form 

a;i  J-i  +  ....  +  a^ii -^u^  0,  [xi-  .  .  .x^  scalars] 

If  independent,  the  motors  XiA^-\-  ....  +a-„J.„^2x.4  form  a  complex  of 
order  n,  called  the  complex  of  A^.  .  .  .  A^.  The  complex  of  highest  order  is 
the  sixth,  to  which  all  motors  belong. 

Two  motors  A^,  A.^,  are  recijyrocal  if  sA^  A^  =  0.  The  ?i  motors  A^  .  .  .  .  A„ 
are  co-reciprocal  if  every  pair  is  a  reciprocal  pair ;  in  such  case  A^  is  reciprocal 
to  every  motor  of  the  complex  A^-  ■  ■  -A,,,  and  every  motor  of  the  complex 
Ai-  .  .  .Ar  to  every  one  of  the  complex  ^r  +  i-  •  •  --^u-  The  only  self-reciprocal 
motors  are  lators  and  rotors.  Of  six  independent  co-reciprocal  motors  none 
is  a  lator  or  a  rotor. 

386.  Theorem.    If  .4,  B,  Cave  motors,  S.ABC=0  if  and  only  if 

(1)  Two  independent  motors  of  the  complex  A,  B,  G  are  lators,  or 

(2)  XA  +  YB  -\-  ZC  =  0,   where  X,    Y,   Z  are  scalar  octonions 

whose  ordinary  scalar  parts  are  not 
all  zero. 

387.  For  linear  octonion  functions  and  octonion  differentiation  reference 
may  be  made  to  M'Aulay's  text.* 

'M'Aii.AY  a. 


TRIQUATERNIONS  AND  QUADRIQUATERNIONS  91 

XVni.     TRIQUATERNIONS  AND  QUADRIQUATERNIONS. 

388.  Definition.    Triquaternions  is  an    algebra   whicli  is    llie    product  of 
quaternions  and  the  algebra' 

389.  Definition.    If  /■  =  ?/•  +  p  +  u  («•,  +  pj  +  ^  {n-.,  +  ^■,)  =  q  -\-  i^qi  -\-  ^q.^, 
where  q,  qi,  q^  are  ordinary  quaternions,  then  we  write  and  define 

r  ■=■  w  +  (cjJi'i  +  jupa)  +  {^w->  +  p  +  cjpi)  =■  G  .  r  -{-  L  .  r  -\-  P.r=zw+l-\-p 

where 

G  .  r  =  w  =  S .q 

L  .  r  =  nio.,  +  p  +  opi  =  UfSq.,  +  Vq  -\-  oS(/i,  called  a  linear  element ; 

P  .  r  =  uwi  +  np.^  =  (dSqi  -f-  ^  Vq-i,  called  a  plane. 

Further,  we  write 

L  .r=:{^  w.,  +  o/?)  +  (p  -f  opi  —  u^3) 

where  we  determine  /3  by  the  equation 

{vM  —  p")  /3  =  wl  p  +  tvo  Fppi  —  p«Sppi 
then  we  define 

m  =  (^  ir.,  +  (j3),  called  a  j^oint 
fZ  =  (p  +  wpi  —  oi3),  called  a  line 
L  .r  =  m  -\-  d 

We  define  further 

L  .  r  ^m  —  d,  the  conjugate  oi  L  .  r 

390.  Theorem. 

G.IV  =G.l'l  LAV   =—L.Vl  P  .IV   —      P.  VI 

G  .Ip    =0  L.lp    =      L.pl  P  .Ip  =-P  .pi 

G .pp'  =  G  .p'p  L  .pp'  =  —L  .p'p  P.pp'=.      0 

391.  Theorem. 

G  .  md  ^  G  .  dm  =  0  P  .  md  =  P  .  dm  L  .  md  =  —  L  .  dm  =  0 

392.  Theorem.    Lr  .  Lr  =  m"  —  d^  l'^  =      ..^    ,,T 

m~  —  d- 


393.  Definitions.    T .  r  =  \^  ic"  ^  U  -  f 

If 

Yq.  =  0  P  .r  =  c^Sqi  =  u?  .  r 

or 

P  .  r=Sqi,   if   Vq,  =  0 

394.  Theorem.    Let 

A  =■  w-  +  ll  —  p-  =z  q~j  +  q.,  q., 

B  =  2{w  Tm  —  TLpd)  =  qq.,  +  qzq 
then 

r-i  =  {A-  -  B-)  {{A  —  fiB)  {q  +  /u7,)  —  0)  (^  -  2,)  q,(q+  q.^)] 
■COMBEBIAC  2.     This  reference  applies  to  the  foUowing  sections. 


92  SYNOPSIS  OF  LINEAR  ASSOCIATIVE  ALGEBRA 

395.  Definition.    Let 

m  ^  (ixq  +  ap  77?'  =  (ix'o  +  op'  c  =  a  +  (j/3 

Then  we  define 

F .  m,  m'  =  o-Q  p'  —  3*0  p  +  oZpp' 
/S  .  c,  m  =  fi  {a-Q^  -f-  G^ap)  +  idGiSp 
S  .  m,  m',  m"=  jS{V  .  m,  m')  m"  =i  S  .  mV .  m',  m" 

396.  Theorem.    V .  wp,  op'  =  oLpp'  /S  .  op,  op'  =■  oGpp' 

S  .  c,  m=:  S  .  m,  c  V .  m,  m'  =  —  Vm',  m 

G  .  c  S  .  c,  m  =■  0      L  .  c  S  .c,  m  =1  iPc^.  tti  P  .  c  S  .  c,  in  =  S  .  c,  Lcm 

G  .  S .  c,  m  .  m  ■=  0    L  .  S  .c,m  .VI  :=V.  in,  Lcm    P.S  .c,m  .  m  =  0 
G  .  ni  V.  m,  m'^  0      Z  .  77i  Vm,  m'=  0  P.  mV.  m,  m'^ — *S'.  m,Lm  m' 

397.  Theorem. 

GW  =  a-o  a;^  +  %' 

Lir  =  7pp'  +  o  [  V{pp{  +  p,  p')  +  Xq  Pi  —  xo  pi] 

PW  =  II  {x,  p'  +  x^  p)  +  o>S'(ppl  +  pi  p') 

i77l  77l'  =  a^o  Pi ^0  Pi 

398.  Theorem.    Gpp'  =—  Tp  Tp'  cos  {p,p')  Lpp'=  Tp  Tp' h  sin  {p,p') 

399.  Theorem.    I  =■  ^Xq  4-  p  +  opi  p  =  «a  +  oi«  Glp  =  0 

Lip  =:  jU/Spa  +  a:o  a  +  o  (T^jp  -\-  Vpi  a) 
Let  ^,  o,  o'  be  units  satisfying  the  multiplication  taVjle 


^ 

o 

o' 

o' 

1 

o 
— o' 

—  o 
0 
-2(^+1) 

1 

(J 

2(^-1) 

0 

and  let  the  quadriquaternion^  A  be  defined  bj  the  equation 

A  =  q  +  (xqi  +  aqo  +  Jq^ 

where  q,  q^,  q.i,  q^  are  real  quaternions.     The  units  i.i,  o,  o'  are  commutative 
with  q,  ji,  q.,,  q^.     If  ^'3  =  0,  A  becomes  a  triquaternion. 

We  may  write  A  as  the  sum  of  three  parts  each  of  which  may  be  found 
uniquely  : 

A=G.A  +  L.A+P.A 
where 

G.A  =  S.q 

L  .  A=  F.  (7  +  //iS'.  <7,  +  o  F.  q.  +  o'  F.  (73 

P  .A  =  (U  F .  (/i  +  oA\  5-0  +  o'*S' .  5:5 

Then  the  formulas  of  §.390  above  hold  for  quadriquaternions  as  well  as  for 
triquaternions,  if  ^  ^  Z  .  A,  p  =  P .  A,  etc. 

'  COMHKlilAC  3. 


SYLVESTER  ALGEBRAS  93 

XIX.     SYLVESTER  ALGEBRAS. 

1.     NONIONS. 

400.  Definition.    Nonions  is  the  quadrate  algebra  of  order  9,  corresponding 
to  quaternion-s,  which  is  of  order  4.     In  one  form  it.s  units  are* 

''-no       ^\:lO       ^IM      \'10       ^-^M       ^iM       ^:m       ^^820       ^^330 

401.  Tlieorem.    The  nonion  units  may  be  taken  in  tlie  forms  (irrational 
transformation  in  terms  of  co,  a  primitive  cube  root  of  unity) 

^0  — -  1  — -  '^iin  T    ^^220  +    ^x]o    ^   ^^  ^no  T  <^^220  i"  '^'''''•aio      *  — ^  '^iio  4"  'J"''>-220  "1"^  ''■330 

J      •—  '^l-.'O  ~r      A^.30  +      XjiQ    ^"  =  Aj3o  +      Aoio  +       /I320       y  ^^  ^^120  "i"  ''-'  '^230  "F'J'^aio 

V  -^  /l];jO  +  oPlojq  +(J"A3:io     fj  =  AjoQ  +0-/l;>3()  +  (')\-i|0     ^7"  ^—  ^130  "H  "'\'I0  ■(""''•320 

whence" 

t^^zi         y3^i         (y?  =  i         (yT=i         (5T=  i         (*Yf=i 

402.  Theorem.    If 

<?  =  2  a-„6  iV"  «,  6  =  0,  1,  2 

Then 

S  .J   <p  =  Xq2  S  .if  (p  =:  I^Xjo  S .  i'j  ^  :=  (J  X12 

S  .y^  =  a-fli  S .  y  "4>  =  "a"2i  S .  i^j'^  =  orxn 

S.^^  =  S{Exa,  iV)  i^Uca i'j")  =  SX X,, y,, a."=  i"+=y*+<^  (?+,'5S)  (,nod  3) 
=  (a-oo  Z/oo  +  a-10  Z/20  +  ^-xi  2/10  +  a^oi  2/02  +  "'a;ji  2/22  +  ua;,!  yjg 
+  a-02  2/oi  +  "^12^/21  +  "°a;o2?/i,) 


Hence 
and  if 
therefore 


^■l  =  2x„j7/,,G)'^i«  +  ''y*  +  ''  a,  J,  c,  fZ=  0,  1,  2 

403.  Definition.  US.j^=0  S.j-^  =  0  S.j=0  then  we 
define 

404.  Theorem.  We  may  write  ^  in  the  form  ^  =  a  +  />*'  +  ci"  (at  least  if 
^  lias  not  equal  roots);  whence,  ify  is  chosen/  so  that  Ay=0,  S.ji^O, 
Sj-i  =  0,  we  have 

A'<^  :=Jcpj-^  z=  a  +  wi/  +  cj" cr  -S'|<|)  =J'^J~"  =  (I  +  (■i'i'i  +  "Ci" 

'  Sylvester  3,  4 ;  Tabek  2  ;  C.  3.  Peirce  6 ;  also  the  linear  vector  operator  in  space  of  tliree  dimensions, 
Bibliographij  of  Quaternions,  in  particular  Hamilton,  Tait,  Jolt,  Shaw  2;   also  articles  on  matrices. 
»  Shaw  7.     This  applies  to  §§403-403.  » Of.  Tabek  2. 


94 


SYNOPSIS  OF  LINEAR  ASSOCIATIVE  ALGEBRA 


405.    Theorem.    If  y  is  to  be  such  that  S.j'  =  0,   Sij'  =  0,  Si"j'  =  0,  and 
if  y  is  such  that  S  .j  =  0,  Sij  =^  0,  Sij^  =  0,   we  may  take 

J'  =  O.J  +  a  J-  +  /3i  ij  +  /?3  if  +  Y\  i'V  +  Y-Jt 
whence 

y-  =  2ai  tto  +  2/?i  73  u-  +  2/3.  yi  CO  +  I  (/?i  aj  +  o'  |8i  ag  +  a^  /?3  w  +  aj  /Jg) 

+  i"  («i  ^2  +  «i  73 "'  +  «3  7i  "  +  0^2  yO  + 

and  if  S .j'''  =  0,  Sij'"  =  0,  ^'ry  =  0  then 

tti  tto  +  u  iSs  7i  +  0)"  /3i  72  =  0         ^i  Uo  +  t'>  «i  f^n  =■  0        tti  73  +  (J  Ua  7i  :=  0 

whence 

ar  =2/^1X1  a^=2/33  73 


and 

That  is 
Hence 
and 


"3  •  1^3  •  ^^3  -—  <^i  •  —  (j"  /^i  :  • —  071  or  a^  =  /ig  =  72  =  0 

J'  =  («!  +  /^i  i  +  7i  i')j=j{(^i  +  "'  /3i  i  +  071  i') 
j,-i  _y3  ^^^  +  /3j  i  +  yi  i2)-i  =  (aj  +  CO-  j3i  i  4  (071  t-)y-i 

y  (a  +  &i  +  ci")j'-^  =j{a  +  bi  +  ci^)j-'^ 


It  is  thus  immaterial  what  vector/  we  take  to  produce  the  conjugate  Kj^, 
except  that  we  cannot  discriminate  between  Kj^  for  one  vector  and  Kj^  for 
another,  if  the  second  is  equivalent  to  the  square  of  the  first.  We  may 
therefore  omit  the  subscript y  and  write  simply  K,  K". 


0 


or 


406.  Theorem.    From  ^  =■  a  +  bi  -\-  ci^  we  have 

<^3  _  3a  ,p"  4-  3  ((t^  —  be)  ^  —  {a^  +  b"^  +  0^  —  3  abc) 

S  .  ^  —  ^       S  .  i^  S  .  i^(p 

S  .  i~^  S  .  <p  —  ^       S  .  i(p  =  0 

S  .  i<^  S  .  i'<p  S  .  <p — <p 

407.  Theorem.^ 

^f\<p  +  ^lC~<p  +  I\(pK''(p  =  3  {S"<p  —  Si^  *S'i-»  =  7;<^ 
^K^K-^  =  S'^  +  SHcf,  +  SH'^  —  3  Sp  Si'^  Sicp  =  T,^ 
T,^=  7\  A>  =  2\  K'<p       7;  4>  =  To  A>  =  7;  A'">        7*3  <?)  =  T's  7v>  =  T,  IC~^ 

408.  Theorem.    If  a  —  1  +  i  +  i",   where  «^  =  Sa-^b  «"/", 

S  .  a^  —  <^  S  .  j~^  a(p  'S' .  y  ~-  a<^ 

.S'  .  aj<p  S  .  y-i  aj<p  —  4)  /S" .  y-2  aj^  =  0 

yS  .  aj'p  /S' . y-'  ay-^  —  ^  S  .  J--  aj~^  —  cp 


iCf.  Tauek  2. 


SYLVESTER  ALGEBRAS 


95 


7;  ^  =  >S'(a  +  ,/-'  aj  +y-2  af)  ^  =  S  (a  +  ICa  +  IC'a)  <p 

—  Aay*?*  'V'^'^^P  —  Aay"</>  >Sj    ''a<lj  —  >SJ'<xj"'^  'V~'''^Jl') 

409.  Theorem.'    <p^  —  SS^  . ^'  +  |  {Z>S~^  —  t^^')  ^  —  (| /S'> 

</'i  ^li  +  <?»i  ^:;  ^'i  +  ^i  ¥i  —  ^'^i  •  (4*1  ^■'.  +  '/'a  <?'i)  —  3'^'<?'::  •  ^1 
+  3  {S-<pi  —  i  SV^^^^)^.,  +  3  ('2A'(^,  A'(^,  —  A'F<^jF^.,)  <^, 
—  (3/S'-<^i  S(p2  +  3A'  F-'(/;,  F^o  —  Z>S<pi  >SV<p^  V^..  —  I  *S'(^.  'S'F=^ij  =  0 

where  V^  =  {^—  S^) 

Also 

4>1  'pi  'P3  +  ^l  <?>;)  ^i  +  <?>:;  <?'l  4>3  +  <p2  4»3  <?>1   +  4*3  ^I  ^'2  +  <?'3  'Pi  ^l 

—  SS^i  .  {(p2^:t  +  'ps'Pi)  —  ^S^j  ■  («?>i4>:)  +  ^.ifpi)  —  3'S'(^;!  .  {^I<p2  +  ^2'Pl) 

+  3A'<^i  (SS(p,  A<^3  —  &'^,  ^,)  +  li^,  (3A'<?)i  Sep,  —  .Scp^  ^,) 
+  3(j);l:iS<pi  S^.,  —  S<pi  ^.,)  —  {27S<pi  S<p..  iS(p:i  —  'JS<pi  >S^..  ^;j 

—  QS<p.,  S^i  ^3  —  SiSi^a  S(pi  ^0  +  '6S(pi  <^o  <^3  +  3  A"!:^!  (^3 1^,)  ^^  0 

410.  Theorem.    If 

/;  {6,  y;)  =  h  [e"  +  "  +  0)^6"'  +  -■-■"  +  0)='''  e^^^  +  ""J  /.•=  0,  1 ,  2 

then 

^  =  {n  ^y  [/o  (^^  ^)  +  i  A  (0,  >7)  +  i'  h  {6, 0] 

=  (7'3<?>)*  [Me,  0)  H-  i-/,  (0,  0)  +  rf,  {0,  0)]  [/o  (^,  0)  +  (A  (>:,  0)  +  r/.  (>:,  0)] 
If  ^1  and  ^i  have  the  same  unit  i, 

^1  =  a  +  bi  -\-  ci-  (^o  =  a'  +  6't  +  di^ 

^i^2  =  {Ts^iy{n^-^'[fo{e  +  e',y!  +  y!')+if,{0  +  6\-r  +  y;')  +  i;%{d  +  e',-^  +  r:)-] 

The  functions/.,  satisfy  the  addition  formulae  ' 
A.(e  +  e',y:  +  y:')=Me,  r)MQ',  rj)  +/,(9,  n)f,^,{&,  rj)  +  /:.(e,  >:)/,+, (6',  rj) 
/,  (u0,  0)  =  «y,  (0,  0)      A  ("0,  co^V)  =  coy,  (0,  >;)      A  (e,  0)  =  y;,  (o,  0) 

K" .  ^=  ( n^y  [/„  (6)'^(j,  <o,7)  +  (/;  {co%  0)^)  +  t-y,  (o-e,  u^)] 

<?,-'  =  (73<?>r*  [/o(-  e,  -  >:)  +  if,  i-d,  —r)  +  i%  {-6,  -r.)] 
r  =  in^r'l /o (po, py;)  +  */i {pO,  p^,)  +  ^y  (;^e, !'>:)] 

411.  Theorem.    Tlie    characteristic   equation    of  <^  =  S  a:„6  i"  j^   may  be 
written 


^00  4-     3-10  +     a-,.,  —  (?) 
Xyo  +  w  a',2  +  o'-Vjo 


+     a-i,  +     a-.j 


•^'o;  "t"      •'■jo  ~r       CToo 


Xuo  +  (J  a-10  +  u"a:"2o  —  ^      -i'oi  +  "3-11  +  u'-'x,! 

a"o3  "I"  ""•'•j2  +  CO  avj  a^oo  "t"  <^'a'iQ  -)-  w  a-oQ  —  ^ 


=  0 


■Taber  2. 


96 


SYNOPSIS  OF  LINEAR  ASSOCIATIVE  ALGEBRA 


The  general  equation  is  tlie  cube  of  this,  but  may  also  be  written 

ABC 


where 


CAB 
B     C    A 


=  0 


A= 


^00  'P       •'^10 


X. 


20 


20 


X 


10 


Xqq  ^       XiQ 

^20  ^00 *?•  ■ 


21 


B=  \ 


6)  J-ji       CJ  J'oi       O  Xn 

o  o  o 


c=- 


2^02 


a-v 


ti)  •^22       ^  "^OS       '^  *^13 

412.  Theorem.  The  cubic  in  ^  has  three  roots,  corresponding  to  which,  in 
general,  there  are  nine  numbers,  in  three  sets  of  three  each,  such  that  each 
set  is  multiplied  by  a  root  of  the  cubic  when  multiplied  by  <^  ;  if  these  are 
pn.  Pi2,  Pis;  p2i,  p22,  P23;  P31,  p32.  p33,  then  Qk  being  the  root  corresponding  to 
the  A;-th  set, 

4>  •  9ki  =  Ok  Pki 

413.  Definition.    The  transverse  of  cp-='Z;r^^i'^p^  as  to  the  ground  defined 

[f  ^  =.  ^j  (p  is  self-transverse. 

414.  Theorem.  We  have  ^<p  =  (p(p,  4*^^^^,  so  that  ^^  and  ^(p  are  self- 
transverse.     Again  <p4^  =^  4'^. 

For 

hence 


Also 
thence 


^4>=^  x„b  Z/,_„,„-b  (o"^-^-'"  i'j-" 


^  =  a-„,  (.-"'*"  y-* 


^^  =  2  a:„,  y,,  uT'^^^'^-'^'i  i«+=y-*-<* 


When 


415.    Theorem.    We  have 

<?)o  =  M<?>  +  ^)  =  i  2  {x,,  +  (."^  x,,.  _,)  i"y* 
i  (<?>  —  ^)  =  i  S  (a:„„  -  o"^  a;„,  _,)  i^y^ 


ab  , 


<?>  =  <?>         a:<jb=  w'"'x„._i, 


1  For  further  theorems  nnd  applications  sec  Joi.Y  1,  2,  3. 


SYLVESTER  ALGEBRAS  97 

2.     SEDENrONS. 

416.  Definition.    Sedenions  is  the  quadrate  of  order  4'.     Its  units  may  be 
expressed  by 

'^•ijo  i,jz=\  ....  4 

It  may  ahso  be  expressed  by  units  of  the  form  ' 

V"  /''  i''  i''' 
where    a,  b,  c,  d  =  0,  1,    and 

^1  *i  "^  ~  ^1  ^1  iz  h  =  —  h  h  h  is  =  iz  h  h  h  =  h  h 

ji  h  =  hj'i  j\  h  =  h  h  i\  =  j\  =  il  =jl  =  —  l 

417.  Theorem.    Sedenions  may  also  be  expressed  in  the  form 

J  =  ^m  +  \':!o  +-  Xaio  +  '^iio  e^,,  =  Cj 

a,b=z0,l,2,3  ji=zs/'^ij 

418.  Theorem.    If 

419.  Definition.    If   ^S'.y=Oand 

S.j^  =  0  =  S.f<p  =  S.f^  y*=l 

then  we  define  '' 

420.  Theorem.    We  may  write  generally  (that  is,  when  ^  does  not  have 
equal  roots,  and  in  some  cases  when  it  has  equal  roots) 


no 

—  ,•«  ^'h 


I 


Whence 


Accordingly 


^  =  a  +  hi  -{-  cp  -{-  dP 

I\(j)  =  a  +  bd  —  ft-  —  dii 
IP^  =  a  —  bi  +  ci-  —  di 
IPcp  =  a  —  hii  —  eP  +  dd 


3 
3 

'3 


bi  =^{^-—  iK^  —  K-^  +  tA' » 

di^  =  i  (<?)  +  iK<p  —  K"-^  —  iK^^) 

421.  Theorem.    Theorems  entirely  analogous    to    those    for   nonions  (see 
§§402-415)  may   be  written   out. 

422.  Definition.    The  transverse  of  0  =  ^x^^  i"/'^,  as  to  the  ground  defined 
^y  *')/»  i«  defined  to  be 


'  Sylvester  3;  Taber  3  ;  C.  S.  Peirce  6  ;  Shaw  8.  icf.  Taber  3. 


98 


SYNOPSIS  OF  LINEAR  ASSOCIATIVE  ALGEBRA 


3.     MATRICES    AS    QUADRATES. 

423.  Definition.  A  matrix  (as  understood  here)  is  a  quadrate  of  any  order; 
that  is,  a  Sylvester  algebra,  usually  of  order  >•  4°.  Its  units  are  called  vids' 
if  they  take  the  form 

\j<s  i,  y  =  1 n 

424.  Theorem.    The  general  quadrate  may  be  defined  by  the  r  ^=  n~  units 

Cab  a,  6  =  1 n 


such  that 
where 


e„b  =  *■"/' 


s=l 

ji  =  ijij 


in  —  jn  ^  l^la  jhyx  —  ^ 


2n 


271 


(J  =  COS 1-  V  —  1  sin  — 

n  n 


If 


a,  h=  I  .  .  .   n 

c,  c£  =  1 .    .  .  n 

a,b,  c,d  ^  1  .  .  .  .n 


425.  Theorem.^   S  .^  =  x^  S  .j'^  (-"  ^  =  x„,, 

426.  Theorem.  Since  every  quadrate  in  the  second  form  may  be  reduced 
to  the  first  form,  it  is  easily  seen  that  ^  satisfies  the  identity  (characteristic 
equation) 


2(0' 


(n-l) 


2  (/<"-"  iCsa 2  u"*"-"  x,o  —  4) 


=  0 


n— 1 


in  each  term  *  2  represents  2 


8=0 


427.    Theorem.    We  may  write 

,^  =  2  i^f  Sj-^'  i-"  ^ 

If  a  =  1  +  t  +  i'^  +  •  •  •    -f  i"~^,  then  the  identical  equation  is 

Saq>  -  <?)  Sj-'  a  4>  ^y-<"-i'  a  ^ 

Saj^  Sj-'aj^-^ Sj-  (»-»  ay  ^ 

Saj''-^^  Sj-^  aj"-'  ^ /S;*-("-"ay<"-''  <?)  —  4> 


a,  b:=  0  .  .  .    n — 1 
a,  b  =  0  .  .  .  .n  —  1 


=  0 


>Laguekre1;  Cayi.etS;  B.  PeibceS;  C.  8.  Peirce,  4,  8 ;  Stephanos  1  ;  Tabeh  1  ;  SiiawT;  Lau- 
BENT  1,  8,  3,  4.     Ou  the  general  topic  see  Bibliograplty  of  Quaternions. 

'SUAW  7;    Lauuent  1.  'Tabek  I!. 

«Cayi.ey3;  Laoukuhe  1  ;  Fuouenids  1,  2;  VVethS;  Tauek  1 ;  Pascu  1  ;  Bucuheim  3;  MOLIEN  1  ; 
Sylvestek  1  ;  Siiaw  7;   WniTEHEAD  1,  and  Bibliography  of  Quaternioni. 


SYLVESTER  ALGEBRAS  99 

428.  Theorem,  (p  may  be  resolved  according  to  tlie  preceding  theorem 
along  any  units  of  the  form  given  by  i,j,  as 

If  J  be  such  that 

S.f^  =  0,J"=l  8  =  l....n—l 

then  <p  may  be  written  in  the  form 

(^  =  a-yo  +  2  a-,,,,  i"j''  a=l....n  —  1      h  =  0  .  .  .  .n—  1 

429.  Theorem.  Whatever  number  ^  is,  /?'  ^  /3~'  has  the  same  characteristic 
equation  as  <p.     Hence  if  this  equation  is 

g"  — W!iC"-i4-  m.,^"--  — ±  w„  =  0 

not  only  is  ^  a  solution,  but  equally  ^'^(3~'. 

430.  Definition.    When 

j"=  1  S.j'^=0  s  =  l n  —  1 

we  shall  call  f^j-*=.  K'  .<p   the  tih  conjugate  of  4). 
If  <^  is  in  the  form  of  §428, 

A'' .  <^  =  a:oo  +  2  x,,,,  cj"'  i'^f'  a=l n  —  1;h  =  0 71  —  I 

Hence  K' .  ^  is  the  same  function  ^  of  o'  i  that  ^  is  of  i- 

431.  Theorem.    We  have  at  once 

<p  +  K.  ^  +  K".^  + +  K"-^  .  ^  =  7uS<p  =  m^ 

(^  +  /r.  4)  + y  =  n\S'.<p  =  'E..K''^K'^       s,f  =  0 n  —  1 

and  since 

(?)"  +  (A»-  + =  4)-  4-  7r .  4)--  + =  7i.S.^' 

therefore 

21.K'^K'^  =  n^  S~^  —  7iS^~=2m„  s,t  =  0 m  —  1,  s  :^  t 

Similar  equations  may  be  deduced  easily  for  3!  m^j  and  the  other 
coefficients. 

432.  Theorem.    If  ^  .  ct  =  (/cr,  then 
also  if 

(<?>  -  i/)  ffi  =  (T, (?>— f/V'-'fTi^cTu  {^  —  gYoy  =  o 

then 

{K'^-g)f<y,  =fa,  ....  (K'^  -  g^-^  f  a,  =j%,         {K'^-gYfa,  =  0 

433.  Theorem.  If  the  roots  of  <^  are  such  that  each  latent  factor  {<p — ^,) 
occurs  in  the  characteristic  equation  of  <p  to  order  unity  only,  then  <^  may  be 
written 

^  =  ao  +  aii  + of„_ii"-^ 

'  Cf.  Tabkr  3. 


100 
and 


SYNOPSIS  OF  LINEAH  ASSOCIATIVE  ALGEBRA 


^  .  aj'  =  (fl„  +  «!  +  ...     a„_i)  .  aj' 


-( («-ii 


a„_i)faf 


Hence  the  latent  regions  of  K^^  are  simply  those  of  ^  transposed.  This 
does  not  necessarily  hold  when  the  latent  factors  enter  the  characteristic 
equation  to  higher  powers.  We  might  equally  say  the  roots  of /f'^  are  those 
of  (p  transposed  (cyclically). 

434.  Definition.  The  transverse  of  <p  with  respect  to  the  ground  defined 
by  i,  j  is 


VJ- 


It  is  evident  that  ^  =  ^. 

If  ^9  =  1  we  call  ^  orthogonal.     If  <^  =  <?>  we  call  ^  symmetric  or  self- 
transverse.     If 

<|)  =  2  {Xij  +  V—  1  yij)  ;iyo.  (a;,  y  real) 

and  if  

^  =  2  (a-ij  —  V—  1  yi,)  ;iyo 

then  ^  is  real  if  ^  ^  ^,  unitary  if  ^^  =  1.  hermitian  if  ^  =  ^. 

435.  Theorem.    The   transverse   of  ^4'  is  'i'^-      Consequently  <p^  =.  <^^, 
and  ^<^  =  ^^. 

436.  Theorem.    We  may  write  the  equation  of  <p,  if 

<p  =  a  +  hi  +  ci'  + +Jd"-^ 


So  that 


S  .  <?)— 4)    S  .i^    ....  S  .  i"-~^ 
S.i'^       S  .i^^  ....  S.^  —  ^ 


S.i-'-'^ 
S  .  i^ 


=  0 


Tz^  =  XK"^  K^^  a,b  =  0 n  —  1,    a:^b 

Ts^  =  2/r"4)  K^ip  K'^ 


T„  4)  =  ^A>  A'-^) /f"-!  ^ 

It  follows  that  if  the  characteristic  function  of  t   be  formed,  it  may  be 
written 


^i-- 


or  for 


By  differentiating  this  expression  in  situ  the  characteristic  function  for 
^„  may  be  formed  in  terms  of  4)1  ...  .  ^n.     This  function  will  vanish  for 


C,  z=  9j ^„  =  (|j„ 


^,  =  K'^,....^^  =  K<^„ 


(iz=  1  . . . .  n  —  1 ) 


PEIRCE  ALGEBRAS  101 

XX.     PEIRCE  ALGEBRAS. 

437.  Ill  the  following  lists  of  algebras,  the  canonical  notation  explained 
above  is  used.  In  the  author's  opinion,  it  is  the  simplest  method  of  expres- 
sion. The  subscrifjts  only  of  the  Jl  will  be  given;  thus  (11 1)  +  a  (122)  means 
Xju  +  aX,23.  For  convenient  reference  the  characteristic  equation  is  given. 
The  forms  chosen  as  inequivalent  are  in  many  cases  a  matter  of  personal  taste, 
but  an  attempt  has  been  made  to  base  the  types  upon  the  defining  equations 
of  the  algebra.  The  designation  of  each  algebra  according  to  other  writers* 
is  given. 

The  only  algebra  of  this  type  of  order  one  is  the  idempotent  unit 

ei  =  >7  =  Xj,o=(110) 

438.  Order  2.    Tijpe  ^  {rj,  i):  {x  —  x^  e^f  =  0 

e2  =  (l]0)  ^^  =  (111) 

The  product  of  ^  ==  a-jCj  +  XzCo,  g  =  i/^  e^  +  y.,  e.^  is 
^a  =  ei  {xi y..  +  x., y^)  +  e^  [xo  y^) 
The  algebra  may  be  defined  in  terms  of  any  two  numbers  t,,  ^,  if  ^  ^^  0, 
so  that  we  may  put  a  in  the  form  a  =  x^  +  y^^. 

439.  Order  3.    Type  ^  {r„  i,  i^) :  {x  —  x^  e^f  =  0 

e3  =  (ll0)  e.,=  {\ll)  ei  =  (112) 

The  general  product  is 

^a  =  cj  (a-iys  +  x.jjz  +  x^yy)  +  e.,  {xny^  +  %%)  +  e^  {x^s) 

The  algebra  may  be  defined  in  terms  of  ^,  ^',  ^,  if  ^"  :|:  0,  ^  :|:  0. 

Ti/pe  *  {y;,  i,j):  (x  —  a-g  63)2  =  0 

eg  =  (110) -f  (220)  eg  =  (210)  ei  =  (lll) 

^0  =  fj  (itj  //;,  +  x^yi)  +  62  {Xi y-i  +  xg 2/2)  +  ^3 3:3  Vz  =  o^ 

The  algebra  is  definable  by  any  two  numbers  ^,  a  whose  product  does 
not  vanish.     The  product  of  ^a  may  be  written 

^a  =  GS^+^Sa  —  e3>SXSa 
Hence 

Also  we  may  write  the   algebra  (>;,  ^',  cr'),  where  ^',  a'  are  nilpotents, 

440.  Order  4.    Type^  {71,1, 1^,1^):  (i  —  ar^ej^  =  0 
ei  =  (110)            63  =  (111)            eo  =  (112)            ei=(ll3) 

If  ^  =  /S^  -(-  V'(,  then  the  algebra  is  defined  by 

^,    C~,    ^',    K\    if  ^4  +  0,     V^^O,    (F0^4:0,    (70^:1:0 

'Enumerations  are  given  bj- PiNCHERLE  1  ;  Catlet  8  ;  Study  ],  2,  3,  8 ;  Scheffeks  1,  2,  3;  Peirce3; 
RoUR  1;  Starkweatueu  1,  2;   Hawkes  1,  3,  4. 

-Study  II;  Scheffers  II, ;  Peirce  a.,.  'Study  III;  Scheffers  III, ;  Peikce  Oj. 

<  Study  V;  Scheffers  IIIj.  '  Study  V  ;  Scheffers  IV, ;  Peirce  a,. 

7 


102  SYNOPSIS  OF  LINEAR  ASSOCIATIVE  ALGEBRA 

Type  {ri,  i,  j,  f)  :  {x  —  x^  e^f  =  0 

e,=  (110)  +  (220)       63=  (210)  + a  (122)       e.,  =  (ill)  +  6  (122)       e^  =  (112) 

^a  =  —  S^.Sa  +aS^  +  ^S(y  +  e^  {x.^ y-z  +  ax^i/g  +  b x.^y^) 
or 

F^  .  To-  =  ^1  (3:3^2  +  a  a-3  2/3  +  6  x,  ^3) 
Hence 

F^  .  Va—  Va  .  V^  =  ^a  —  G^  =  he^  (0-3 2/3  —  0:3^2) 

We  have  two  cases  then  :    (l)  when  6=0,  (2)  when  J  :J;  0. 
We  may  determine  e\=-  e^,  from 

{V^f  =  e,{xl  +  a:4) 

When  a  =  0,  this  gives  us  only  one  case  of  (t°=:  e^. 

When  a  ::j:  0,  we  may  talce  4  =  e^  as  well  as  e|  =  e^ ;    whence,  if  a  ^  0 

gg  63  =  0  eg  Co  :=  0 

If  a  -f  0,  we  may  put  «  =  1 

6363  =  0  e.,€^—0 

Finally,  then,  we  have^ 

{vijf){\)  f3  =  (210)  63  =  (111)  61=  (112) 

(>7*i.f)(2)  ^3=  (210)  +  (122)  6'2=(111)  e,  =  (ll2) 

{vijf){^)  ^3  =  (210) +  (122)  a  e.  =  (1 1 1)  +  (122)  e,  =  (112) 

(>7Ur)(4)  e3  =  (210)  ^3  =  (111) +  (122)  ei  =  (112) 

Type  {yi,  i,  j,  ij) :  {x  —  x^  ej"  =  0 

e,=  (ll0)  +  (220)  63  =  (210)  e.  =  (11 1)  —  (231)  ei=(211) 

^a  =  ^1  (.T3  y.,  —  x.>  y-i  +  Xg  ?/,  +  a-^  y-^  +  e.  (a-,  y^  +  x^  ?/o) 

+  ^3  (^3  ?/4  +  a-4  7/3)  +  e^  Xi  i/i 
Defined  ^  by  ^^,  <7,  such  that  ( V^f  =  0  =  ( Fa)' 

Type'{-^,i,j,k):  {x-x,e,f  =  0 

e,=  (110)  +(220)  +  (330)  63  =  (210)  ('3  =  (310)  ^  =  (111) 

V^Va  =  0 
Defined  by  any  three  independent  numt)ers. 

441.    Order  5.     7)/pe '  (>7,  i,  r,  i\  i*)  :  (x—x, e,f  =  0 

6,  =  (110)  e4  =  (lll)  e3=(112)-      ^.=  (113)  ei  =  {U4) 

Definable  by  any  number  '(  for  which  (  F^')'  :^  0. 


'Stcdt  IX  is  O7,  i,J,j')  (3)  If  «',  =  (310) -(111)  +  (c-l)(12ri),  t,  =  (1U)  +  2(132).  Scuepfehs  IV^  is 
tUc  8am3.  Peirce  6,  and  b\  reduce  to  this  form.  Studv  X  and  ScuEt-PEUs  IV^  reduce  to  (3);  Study  XI 
and  SoiiEKFEiis  IV^  reduce  Id  (1);  SoiiEi'i'Kits  [Vj  roduues  to  (4)  if  ;i  =  —  1,  otherwise  it  reduces  to  (S). 

'Stody  XIV;  SOBEFFEKS  IV,;  PEruCE  d,.  aSTODY  XVI;  Soueffkks  IV,. 

«3CUBFFERS  V,;   Peiuce  a,. 


PEIRCE  ALGEBRAS  103 

Type  '  {ri,  i,  j,  f,  f)  :  (a;  —  Xj  e^y  =  0 

ej  =  (ll0)4-(220)  ei  =  (210)  +  a(l23) 

e,  =  (lll)  +  /v(123)  eo  =  (n2)  ei  =  (113) 

(1)  />  :|i  0,  we  may  lake  1  =  1. 

(2)  h  =  0,  we  may  take  a  =  1,  or 

(3)  b  =  0  =  a. 

Type  '  (>7,  i,  h  y ,  r)  •  (a;  —  a-5 e.f  =  0 

65=  (110)  +  (220)  64  =  (210) +  6  (221) +  c  (122) 

e3  =  (lll) +  (?(221)  +  e(122)        e2  =  (21l)        ei  =  (112) 

(1)  e4  =  (210)  e3=(lll)  +  ci(22I) 

(2)  64  =  (210)  e3=(lll)+(i(22l)  +  (122) 

(3)  ^4  =  (210) +  (122)  e,=  {\n) 

(4)  ^4  =  (210) +  (221)  63=  (lll)-(221)  +  e(I22) 

(5)  64=  (210)  4- (122)  63=  (lll)  +  cZ(221)  +  e(122) 

Type  3  (>7,  i,  i?,  j,  f) :  (x  —  x^  e.f  =  0 

^5  =  (110)  +  (220)  +  (330)  ^4  =  (210)  +  (320) 

eg=(310)  63=  (111)  ei  =  (112) 

Tl/pe '  {r;,  i,  j,  h,  !r)  :  (x  -  x,e,f  =  0 

f5=  (110)  +  (220)  +  (330)  64=  (210)  +«  (122)  +  Z-(132) 

e3  =  (310)  +  c(l22)  +  (Z(l32)     e.=  (HI)  +  e  (122) +/(132)     ei=(ll2) 

(1)  64  =  (210)  +  (122)  63=  (310) +  (132)     e,  =(111)    ei  =  (ll2) 

(2)  64  =  (210)  eg  =  (310) +  (132) 

(3)  64  =  (210)  eg  =(310)  

(4)  64=(210)  +  (l22)-y(l32)  e3  =  (310)  +  7(122)  — (132)     ^^=(111) 

(5)  ^4  =  (210) +  (122)  — (132)  C3  =  (310)  +  (122)      

(6)  64=  (210)- (132)  63  =(310)  +  (132)      

(7)  e4  =  (210)  +  (l+a-')(122)  e3=(310)  

(8)  e4  =  (210) +  (122)  e3=:(310)     ej  =  (1 11)  —  2  (122) 

(9)  e4  =  (210)  +  (122)  e3=(310)+  2(122)  — (132) 

e3=(lll)—  2(122) 
(10)      e4  =  (210)  — (122)  +  (132)     63  =  (310)  —  t  (122)  —  (132) 

e,  =  (lll)—  2((132) 

'SCHEFFERS  V,  is  in  (1),  «,  =  (210)  +  (123)  —  (112),  e,  =  (111)  +  2(123);  Schefpers  V,  is  (2);  Schef- 
FERS  V,  is  in  case  (1),  a  =  0,  c,  =  (210)  —  (112) ;  Schefpers  V,  is  (3);  Peirce  6^  is  in  (1),  j=  (111)  —  (123), 
fc  =  (n3),  J  =  (113),  m  =  (210)  +  (133)  +  (112);  Pkirce  c^  is  in  (1),  j  =  (HI)  -  (123),  *  =  (112),  I  =  (113), 
m  =  (210)  +  (112). 

«SCHEFFEKS  V,.  is(l);  e,  =  ( 1 1 1)  +  ?.  (231),  e^=  (210);  V,,  is  (3?  with  d  =  —  1;  V„  is  in  (5);  V„  is  in 
(3)  or  (4);    Peirce  d^  is  in  (5);  e^  is  in  (4);  /^  is  in  (1);  ^5  is  in  (5);  Aj  is  in  (3);  tj  is  in  (1). 

•ScHEPFBRS  V||.;  Peirce  J..  *  These  are  in  order  Schefpers  V^,  —  Vj^. 


104  SYNOPSIS  OF  LINEAR  ASSOCIATIVE  ALGEBRA 

Type  {r„  i,  j,  h,  I):  {x—  x^  eS~  =  0 

(1)1     65=1(110)  +(220) +  (330) +  (440)  c,  =  (210)  — (l3l) 

e3=r(310)  +  (12l)  6.=  (410)  ei=(lll) 

(2)'     ^5=  (110) +  (220) +  (330) +  (440)  e4=(210) 

eg  =  (310)  62  =(410)  ei=(lll) 

442.    Order  6.    Type  ^  {yj,  i,  r,  i?,  i\  i^) :  {x  —  Xa  e^f  =  0 

66=  (110)         65=  (111)         6,  =  (112)         63=  (113)         6,=   (ll4)         61  =  (115) 

Type '  {-r,  i,  j,  f,  f,  f) :  {x-x,  e,f  =  0 

66  =  (110)+ (220)      65=  (210)  +  a  (124)      e^  =  (11 1)  +  &  (124) 
63  =  (112)  63=  (113)  61  =  (114) 

(1)  a=l=b  65=  (210) +  (124)  6,  =  (lll)  +  (124) 

(2)  a  =  0,J  =  l         e5=(210)  64  =  (ill)  +  (124) 

(3)  a  =  0=b  65  =  (210)  64  =  (111) 

Type '  (57,  i,  J,  ij,  f,  f)  {x—x^ e^)'  =  0 

(1)  65=  (210)  +  (122)+  2\/^^(22l)     64  =  (111) +  (221) 

(2)  65  =  (210)  6,  =  (111) +  2(123) 

(3)  6a  =  (210)+  (123)  6^  =  (111)  +  2  (123) 

(4)  65  =  (210)  6,  =  (111) +  6^(221) 

(5)  65=  (210) +  (221)  e4  =  (lll) 

(6)  65  =  (210)  +  (123)  64=(111)  +  (^(221) 

(7)  66=  (210) +(221)  64=  (111) +(123) 

(8)  65  =  (210)  e4=(lll) 

(9)  65  =  (210) +  (123)  64=  (111) 

10)  65  =  (210) +  (122)  e4  =  (lll)  — (221)  — 2(122) 

11)  65  =  (210)  +  (122)  64=(111)  — (221) 

12)  65  =  (210) +  (123)  64  =  (111)— (221)  — 2  (122) 

13)  65  =  (210)  64  =  (111)  — (221)— 2(122) 

14)  65  =  (210)+  2(1  q:  v/:i^)(22l)  +  4\/^=T(l22)+  (123) 

e4  =  (lll)  =F  >/^^^(22l)  +  2(1  ±  \/^^)122 

15)  65  =  (210)  +  2  V-^  (221) +  (122)     64  =  (1 1 1)  +  (221)  +  2  (l  23) 

16)  65=  (210) +  4  (221)  + (123)  64  =  (1 1 1)  +  (221)  +  2  (122) 

17)  65  =  (210) +  4  (221)  64  =  (111)  +  (221)  +  2(122) 

18)  65=  (210)  + 4(221)  +  (123)  64  =  (111)  +  4  (122) 

19)  e5  =  (210)  — (?H  — 1)(221)  — i(???  +  1)  (?/>  —  3)  (122) 

^^^(111)  +  ^'^  3  (221)+  2(122) 

'SCIIEFFERS  V,j  .  'SCHEFFERS  V33.  apEiKCEn,,.  *  Peiuce  ()g  is  ( I ) ;    Cg  is  (3). 

•These  arc  In  order  Staukweathek  4,  8,  9,   U,  13,  13,  14,  15,  16,  I'J,  30,  31,  33,  33,  27,  29,  30,  33,  33. 
Also  Peirce  aoj  and  h)„  are  iu  (4),  nd^  in  (5),  z,.  in  (6),  «/„  in  (8),  ae„  in  (tt),  ?«,.  in  (II). 


PEIRCE  ALGEBRAS  jq^ 

(1)  .',  =  (210) +  (.320)  e,=(.310)    .,  =  (111)    e,  =  (ll2)    «,  =  (113) 

(2)  c'6=(210)  +  (320)  +  (133)      e,=  (,310)  +  ( 1  23)         6,=  (ill) 

e.,  =  (\['l)    .-,  =  (113) 

(3)  ^6  =  (210)  + (320) -f  (133)     e^  =  (310)  +  (123)         6,=  ( 1)  1)  +  2  (123) 

eo={n2)  e,  =  (ll3) 

(4)  e,  =  (210)  +  (320)  e,  =  (;310)  e,=  (1 1 1)  +   2  (123) 

Co  =  (112)  e,  =  (ll3) 

Tljpe '  (r,  i,  j,  k,  /r,  P)  (^  _  ^^^  ^^y  ^  0 

(1)  ^,  =  (210)  e,  =  (310)    e,  =  (lll)    e,  =  (112)    .i  =  (ll3) 

(2)  e,=(210)  +  (123)    e,  =  (310)    e,,=  (lll)    e,  =  (ll2)    .,  =  (113) 

(3)  .a=(210)  e,  =  (310)  e3  =  (lll)  +  2(123) 

«3  =  (ll-3)  ei  =  (ll3) 

(4)  e,  =  (210)-(133)  e,  =  (3I0)  +  (l23)  e3  =  (lll) 
.                                                              e3=(ll2)  e,  =  (1)3) 

(^)     ^«=(210)  e,  =  (310)  e3  =  (lll)  +  2(133) 

^^=(112)  e,  =  (ll3) 

(6)  e'5  =  (210)  +  r/(l33)                   e,  =  (310)  +  (123)  e3  =  (lll) 
^   ^                                                             e,  =  (112)  ei  =  (ll3) 

(7)  e,  =  (210)  +  (133)  e,  =  (310)  +  (123)  ^3  =  (l  1 1)  +  2(123) 

^2=(112)  ei  =  (ll3) 

(8)  e,  =  (210) +  (133) +  (123)       e,  =  (310)  +  (123)  .3  =  (1  H)  +  2(133) 

^2=  (112)  ei  =  (ll3) 

'^m  iv,  i,j,  ij,  f,  if)  C.3  =  1  (a;  _  ^^e,f  =  0 

e«  =  (210)  +  ^(l_co)(22l)-|<.(l22)     e,  =  (111)  +  c.  (221)-^  (1-.,)  (1  22) 
e3=(21l)  +  i(l_,,)(222)  e2  =  (112)  +  .,r(222)  e,  =  (212) 

Tl/pe  (>:,  i,  J,  k,  ik,  V')  (^  _  ^^  ^^y  ^  0 

(1)  e5  =  (210)  e,=  (310)  +  (132)  ^3=  (111)  +  6(i22)  +  c  (132) 

62  =  (211)  ei  =  (112) 

(2)  e,=  (210)  +  (l22)       .^  =  (310)  +  (132)       ^3  =  (HI)  +  6(122)  +  c  (132) 

e2  =  (21l)  e,  =(112) 

(3)  e6  =  (210)+a(122)     e^  =  (310)  +  (122)  +  (132) 

e3  =  (lll)  +  6(122)  +  c(l32)  e.  =  (211)         e,  =  (112) 

(4)  e5=(210)  e,=  (310) +  (122) +(132)      e,  =  (1 1 1)  +  i  (122)  +  c(l32) 

^3  =  (2n)  f,=  (ll2) 

(5       e6  =  (210)  +  (l32)  ,,  =  (310)-(122)  e3=(lll)  ,,=  (211) 

(6)  e5  =  (210)     e,  =  (310)  6>3  =  (lll)  +  (i22)  e,  =  (211)  e;=(ll2) 

(7)  ^,  =  (210)     e,  =  (310)  e3  =  (lll)  .,  =  (211)  e,  =  (112) 

'These  are  in  order  Starkweather  3,  5,  28,  10. 

'  These  are  in  order  Starkweatheu  1,  2,  0,  17,  18,  24    2.^    26. 


106 


SYNOPSIS  OF  LINEAR  ASSOCIATIVE  ya,GEBRA 


(1) 

(2) 
(3) 
(4) 
(5) 
(6) 
(7) 

(8) 

(9) 
(10) 


(1) 

(2) 
(3) 
(4) 
(5) 
(6) 
(7) 

(8) 

(9) 
(10) 


Type{r,i,j,h,l,l') 
f5  =  (2l0)-a(132) 


66=  (210) 


{X- 

e^=  (310)  + a  (122) 

e2  =  (lll)  +  a  (122)  +  fl(132)  +  a(142) 

eo  =  (111)  + a  (122)  + a  (132) 

<'3=(111)  +  a(122)  +  a(142) 

62=  (111)  +  o(132) 

62=  (111)  +  a(142) 

e2  =  (lll) 

e^=(310) 

63=  (111)  +  (122)  +  (132)  +  (142) 

eo  =  (lll)+  (122)  +  (132) 

e2  =  (lll)  +  (122) 

Co  =  (111) 


a-eCs, 


Type  {yi,  i,  j,  k,  J,  il) 
e5=(210)  — (231) 


66  =  (210) 


{x. 


Type{Yi,  i,j,  k,  I,  m) 


e,=  (310)  +  (22l) 

e2  =  (lll)+(221)+(23l)  +  (24l) 

6.  =  (111)+  (221)  +  (231) 

e.,=  (111)  +  (221) +  (241) 

62=  (111) +(221) 

62  =  (111)  +  (241) 

e2  =  (lll) 

e^  =  (3l0) 

62=  (111) +  (221)  +  (231)  +  ^241) 

62  =  (111) +  (221) +  (231) 

62=  (111) +  (221) 

62=  (111) 


y  =  o 

e3  =  (410) 
6,  =  (112) 


63  =  (410) 
61  =  (112) 


•^6^6. 


f3=(410) 
61  =  (211) 


e3=(410) 

6i  =  (21]) 


{x- 


C6=(210) 


64  =  (310) 


63  =(410) 


62  =  (510) 


a*6  e^f  =  0 

6i  =  (lll) 


SCHEFFERS  ALGEBRAS  107 


XXI.     SCHEFFERS  ALGEBRAS. 


443.  Tlie  following  liwls  include  algebras  of  order  less  than  seven,  with 
more  than  one  idempotcnt.  lieducible  algebras  aie  not  included,  nor  are 
reciprocal  algebras  both  given.'  'J'he  idempotents  are  >;;  direct  units  t,  j. . . .; 
skew  units  e. 

444.  Order  3.    Type"  (>7i ;  r,.. ;  e.,^)  {x  —  x.^  e^)  {x  —  x-^  Cq)  =  0 

63  =  (110)  6.  =  (220)  ej  =  (210) 

445.  Order  4.    'Type  '  {yj^ ;  j?.,  i;  ejj)  (a;  —  a-;,  e^)  {x  —  x^  e^f  =  0 

^^  =  (220)  ey=(110)  e.  =  (lll)  ei  =  (2l0) 

Type  M>7i ;  >72 ;  Cai ,  e'21)  {x  —  x-i  Co)  {x  —  X4  e^))  =  0 

e^  =  (110)  e;j  =  (220)  <?.  =  (210)         ei  =  (21l) 

Type  ^  (>7j ;  y/., ;  % ,  e,.)  (x  —  x.j  «„)  (x  —  x^  gy)  =  ^ 

e,  =  (110)  63  =  (220)  Co  =(121)  ej  =  (2ll) 

446.  Order  5.    Type  ^  {1^1 ,  i,  i~;  ■^,,  e^^)  (x  —  Xj  ?„)  (x  —  X;  Cq)'  =  0 
6-6=  (110)       e,  =  (220)       e,=  (lll)       e..,  =  {\\i)      e^  =  (211) 

Type ''  {y:i,i;  ^i,j;  eoi)  (^  —  x^ eo)^  (x  —  xg eo)2  =  0 

65  =(110)       ^^  =  (220)        e,  =  (111)      e2  =  (222)       €^  =  (211) 

Type  M>7i ,  * ;  >72  ;  ^^i ,  e^i)  (x  —  Xi  Co)  (x  —  X5  ej-  =  0 

(l)e6=(110)                   e,=  (220)  e.  =  (lll)     e,  =  (211)     ^^  =  (212) 

(2)e5  =  (110)                   6i  =  (220)+(330)  e,=  (lll)     e.  =  (21l)     ei=(310) 

(3)   65=  (110)+(220)     e,=  (330)  e3  =  (210)     £,  =  (111)     ej  =  (311) 

Type  ^yiifi;  n-i)  ^12,  e,^)  (x  —  x^  e^)  (x  —  x^  e^f  =  0 

(l)e6  =  (110)  e,  =  (220)         e3=(122)  e^  =  (210)         ^^=(112) 

(2)e6=(ll0)         e,  =  (220)         e3=(122)         &,  =  (211)         ei=(112) 

Type  ^"  {m ;  >72 ;  4,  4',  e^i")  (x  —  x^  e,)  (x  —  Xi  Co)  =  0 

66=  (110)      ei  =  (220)+  (330)4- (440)       e,  =  (211)      e,  =  (310)      £^  =  (410) 
Type  "  (>7j  ;  >7o ;  e^o,  el^,  e'.J,)  (x  —  x^  Co)  («  —  ^i  Co)  =  0 

e6=(llO)     e^  =  (220)  +  (330)     63=  (121)      e.  =  (211)     ei  =  (310) 

Type  ^2  ()7i  J  Yi.,;  yi-y,  e,^ ,  e,^)  (x  —  X3  e,)  (x  —  x^  e^)  (x  —  x,  ej  =  0 

65  =  (110)       ei=(220)       63  =(330)      e»=(221)      ^^  =  (311) 

Type '^  (>7i ;  )73 ;  yi-^;  e.^,  e-^)  {x  —  X3 Cq)  (x  —  x^  e^)  (x  —  Xj e^)  =  0 

eg  =  (110)       ei  =  (220)       63  =  (330)      63=  (211)       ei  =  (321) 

'For  algebras  of  order  seveu  see  Ha WKE3  4.  "These     are    in    order    Scheffsks      V,,,    V„,    V,,  • 
•StUDT  IV;  SCHEFFEKS  III, .  Hawkes  (V)3„,  3,j,  1,. 

•  Studi  VII;  SCHEFFEKS  IV,.  'These  are  in  order ScHEFFEBS  V„,  V,,;    HawkE3(V)3     3. 

< Study  XV;  Scheffeus  IV,.  '»Scheffebs  V„;  Hawkes  (V)  5. 

tSTUDT  XIII;   SCHEFFEBS  IV,.  >' ScUEFFEKS  V„ ;  HaWKBS  (V)  6. 

•SCHEFFEBS  V,;    HaWKES(V)1,.  "  ScDEFFEKS  V^. 

'SCHEFFEBS  V,;    HaWKES  (V)  4.  "SCBBFFEBS  V,. 


108  SYNOPSIS  OF  LINEAR  ASSOCIATIVE  AX^GEBRA 

447.    Order  6.    Type^  {r,^,  i,  t\  P;  r.^;  e^-^)  {x  —  x^e^)  {x—Xze^^=  0 

eg  =  (110)        e5  =  (220)        e^  =  (221)        e,=  {222)        e.=  (223)         .'j  =  (210) 

Type 2  (>:i ,  ii ,  j\ ,  jl ;  r,. ;  e.^)  {x  —  a-j  ?„)  (a-  —  x^  e^f  =  0 

(1)  eg  =  (220) +  (330)        e,=  {\W)         e,  =  (320)        63^(221)         e.=  (222) 

ei=(212) 

(2)  e,  =  (220)  +  (330)        65=1(110)        e,  =  (320)  +  (232)  e3=(221) 

63=  (222)  ei=(212) 

(3)  66=  (220) +  (330)        e6=(110)        e^  =  (320)  +  a(232) 

63  =  (221)  +  (232)  e.  =  (222)        ej  =  (212) 

(4)  eg  =(220)  + (330)        e5=(110)        e,  =  (320)        63  =  (221)  +  (232) 

e3  =  (222)  ei  =  (212) 

Type  ^  (>:i ,  ii ,  j\ ,  ii  jl ;  n. ;  e^i)  {x  —  Xg  eo)  {x  —  x^  e^f  =  0 

66=  (110)      eg  =  (220) +(330)      e^  =  (221)  —  (331)      63  =(320)       e.  =  (321) 

e,  =  {2\\) 

Type  ^  {y:i,h,  j\ ,  h  ;  r,.^ ;  Cgi)  (^  —  ^^  ^0)  (•»  —  -^'s  <?o)"  =  0 

eg  =  (110)       eg  =(220) +  (330)  + (440)       e,=  (320)      63  =(420)      60=  (221) 

ei=(211) 

Type ^  {rn,  h ,  iV'  ^2,  h;  (^21)  i^—  ^5  e^f  {x  —  x^  e^f  =  0 

e6=(220)       €,  =  {\\Q)       6,  =  (221)        e^=  {\\l)       e,=  (112),        e,  =  (122) 

Type  ^  (J7i ,  I'l ,  ji ;  573 ,  i-i ;  e.^)  {x  —  x^  e^f  [x  —  x^  e^f  =  0 

e6=(330)    ^^  =  (110)  +  (220)    e4  =  (331)    e3=(210)     e3=(lll)      e,  =  (31l) 

Type''  (>7,  ti,  ij;  y-.^ ;  e^^,  gj,)  [x  —  x^  ep)  (x  —  Xg  e^f  —  0 

(1)  e6  =  (llO)       65=  (220)+  (330) +  (440)  e,  =  (221)  +  (430)       e3=(222) 

6,  =  (310)  61  =  (410) 

(2)e6  =  (ll0)       es=  (220) +  (330)  +(440)  ei  =  (22])  e3  =  (222) 

62=  (310)  61  =  (410) 

Tyjje  ^  ()7i ,  ^1 ,  q ;  >72 ;  ^12 ,  e^i)  (x  —  Xg  e,)  (x  —  Xj  Co)^  =  0 

(1)  eg  =(330)  +  (440)    e^  =  (110)  +  (220)    £',=  (132)    63  =  (310)    e.  =  (lll) 

6,  =  (112) 

(2)    e,  =  (142)      

Type ^  {y;i,h,  j\ ;  >:,. ;  Co, ,  ei^)  (x  —  xg  t-o)  (x  —  xg co)'-  =  0 

(1)^6  =  (110)     65  =  (220)  +  (330) +  (440)  +  (550)     e,  =  (320)  +  (540) 

63  =  (221)      62  =(410)         p,  =  (510) 
(2)    64=  (320)  

UIawkes  (VI)  1,  1.  <  Hawkes  (VI)  1,  6.  MIawkes  (VI)  3,  1,  3,  2. 

=  In  order  Hawkes  (VI)  1,  3,  1,  4,  1,  2,  .     '  Hawkes  (VI)  2<  1.  »  Hawkes  (VI)  4^  1,  4,  3. 

aiUwKES  (VI)  1,  5.  'IlAWKES  (VI)  3,  3.  «IlAWKES(VI)  3^  3;  3,4. 


SCHEPFERS  ALGEBRAS  109 

Type '  (>:, ,  i, ,  y, ;  r,., ;  e.,, ,  e,„)  (x  —  Xj  t'u)  {x  —  aij  Cq)^  =  0 

(1)  e„  =  (330)  +  (440)       e,  =  (110)4-(220)  +  (550j       ^',  =  (131)       e^  =  {^\0) 

e.,  =  (210)       ei=:(lll) 

(2)    e,  =  (141)       e^^l^ill)         

^!'/JPe "  (>7i .  *i  i  >:3,  *2 ;  ''12,  eiO  (^  —  a^6  eo)'  (a^  —  ^s  eo)'  =  0 

(1)  ee  =  (440)     rv,  =  (110  +  (220)  +  (330)      e^  =  {\i\)     e3  =  (441)     e,=  (i40j 

e,  =  (14l) 
(2)e„  =  (440)     e5  =  (110)  +  (220)  e4  =  (lll)     e,=  (441)     «?„  =  (240) 

ei  =  (:i41) 
(3)e„  =  (440)     e5  =  (110)  +  (220)  +  (330)    e,=:(lll)     63^(441)     ^,  =  (340) 

e^  =  (240j 

Type '  (m ,  ''i ;  r,., ,  u ;  e,o ,  Cji)  (x  —  x,  e„)^  (x  —  x^  e^f  =  0 

(1)  ^8=  (330) +  (440)  e,=  (110)  +  (220)  e^  =  (310)  +  (421) 

63  =  (131) +  (240)        e,  =  (441)        €^  =  (111) 

(2)      e3  =  (240)  

(3)      e,  =  (310)     63=  (131) +  (240)         

(4)      e3=(240)  

Type^  (>7i,  i\ ;  r,.,;  e[.,  e^l,  e'l^)  (x—  x^eo)  (x  —  Xjeo)'  =  0 

(1)  66  =  (440)      6,  =  (ll0)  +  (220)  +  (330)  e,  =  {in)      63  =  (340) 

e,  =  (140)      61  =  (141) 

(2)  6e=(550)      65=  (110) +  (220)  +(330) +  (440)      e4  =  (lll)      63  =  (150) 

6,  =  (250)      61  =  (151) 

(3)      63  =(350) 

ej  =  (450) 

Type ^  iyii,h;yi2;  e'n,  ^{2 ,  eg,)  (x  —  Xg Cq)  (x  -  Xg eo)'  =  0 

(1)  66  =  (440)  +  (550j      65  =  (110) +  (220) +  (330)      e^  =  (530)     63  =  (140) 

fe,=  (lll)     6,  =  (141) 

(2)     e3=(141) 

61=  (240) 

Type  ^  (>7i ,  J'l ;  )72 ;  ^21 ,  eii,  eij)  (x  —  x„  60)  (a:  —  Xg 60)'  =  0 

66  =(440) +  (550)  65  =  (110) +  (220) +  330)  6,=  (410)  63=  (141) 

62  =  (111)  61=  (530) 

Type '  {r;i ;  y;^ ;  6,'.,,  e;{ ,  ei'.i',  ej-J)  (x  —  Xg  60)  (x  —  Xg  69)  =  0 

e^  =  (660)    65=(110)+  (220)  +  (330)  +  (440)  +  (550)    e^  =  (460)    e^  =  (360) 

60  =  (260)    61=  (160) 

'  Hawkbs  4,  3,  4,  4.  *  Hawkes  (VI)  5,  3,  7,  ],  7,  3.  «  Hawkes  (VI)  8,  3. 

sHawkes  (VI)  5,  1,  5,  3,  5,  4.  '  Hawkes  (VI)  8,  1,  8,  3.  i  Hawkes  (VI)  9, . 

«  Hawkes  (VI)  6,  1,  0,  2,  G,  3,  C,  4. 


» 


110  SYNOPSIS  OF  LINEAR  ASSOCIATIVE  ALGEBRA 

%)e  1  {ra ;  >:2  i  «i2 ,  «)2,  Cjo',  %)  (a^  —  JCs ^o)  (a:  —  a^B  ^o)  =  0 

^5=  (110) +(220) +(330)  +  (440)    65=  (550)+(GC0)    e,=  (4G0)    e3=(360j 

62=  (260)    ei=  (510) 

!%7e  2  (>7i ;  >72 ;  Cjo  e,'^  ej  e^j)  (a;  —  Xg  e^  {x  —  a:^  fo)  =  0 

e,=  (440) +  (550) +  (660)  e^  =  (1 10)  +  (220)+ (330)  e4=(630) 

63  =  (530)  e.  =  (250)  e^  =  (140) 

%9e  ^  ()7i ;  ii ;  yi.;  573 ;  e^o,  Cig)  (a:  —  a^^  eo)  (a:  —  a^s  ^o)  (a;  —  arg  e^f  =  0 

ee=(110)       e,  =  (220)       e,  =  (330)       e3=(313)       ^.=  (323)         Cj  =  (333) 

^l/F^ *  (>7i >  *i ;  ^"~'j  ^s->  ^2] ,  ^23)  (a:  —  a;4 Co)  (x  —  x^ e,,)  (a;  —  a-^ Co)'  =  0 

ee=(110)       e,  =  (220)       e,=  (330)       f3  =  (212)       e2=(232)         ei  =  (33l) 

%'e  ^  ()7i,  ii ;  >73 ;  >73 ;  ^13,  eo,)  (a;  —  a-^  Cq)  (a:  —  ^5  Cq)  (a:  —  a'c  Cq)"  =  0 

66  =  (110)       eg  =(220)       64  =  (330)       63  =  (231)       6,=  (312)         ei=(332) 

Ti/pe  ^  (>7i ,  ii ;  yjo]  vja]  e.i ,  632)  (a:  —  ^i  «o)  (a:  —  scr,  Co)  (a:  —  a-j  fo)'  =  0 

e,  =  (110)       e5=(220)       e,  =  (330)       e3=(12l)       e2  =  (23l)         ei=(331) 

Ty2>e  •  {y-i, ;  r,.. ;  ^3 ;  e:2,  ^is,  ^ss)  (a^  —  ^i  ^o)  (a:  —  arg  Co)  (a:  —  a^e  <°o)  =  0 

ee=(llO)     65=  (220)     e,  =  (330) +  (440)    e3=(31l)     e.^  =  {A20)     e,  =  {Z2\) 

Type ^  ()7i ;  773 ;  r,^;  e^. ,  e^^ ,  e^)  {x  —  x^ e^  {x  —  x^ e^  (x  —  x^ e^  =  0 

(1)  ee  =  (ll0)      e5=(220)      e,  =  (330)      e3=(312)      e2=(23l)       ei  =  (322) 

(2) e3  =  (211)      63  =  (320)       ^^=(311) 

(3) e2  =  (32l)        ei  =  (31l) 

Type ^  (>7i ;  yi-i\  »?3  i  «i2,  ^12 ,  ^si)  (a:  —  a-^  e^  (x  —  x^  e^  {x  —  x^  e^  =  0 

65  =  (110)     65  =(220)     e,  =  (330)  +  (440)     63  =  (420)    6.=  (130)      ei  =  (321) 

Type  1"  (»:i ;  r,.;  ris]  e^,,  e.^,  631)  (x  -  Xj  eo)  (x  —  Xg  e^)  (x  —  Xj  Co)  =  0 

e,  =  (ll0)       eg  =(220)       e,=  (330)       63  =  (211)       e2  =  (13l)        Cj  =  (321) 

1  Hawkes  (VI)  10^.  '  Ha WKES  (VI)  3,  3.  «  Hawkes  (VI)  4, ,  83  1,  9,  3. 

SHaTVKES  (VI)  11,.  6HAWKES  (VI)  73.  »  Hawkes  (VI)  Sj  . 

3  Hawkes  (VI)  Is  3.  '  Hawkes  (VI)  3,.  '"Hawkes  (VI)  83. 
<  Hawkes  (VI)  63. 


CARTAN  ALGEBRAS 


111 


XXn.     CARTAN  ALGEBRAS. 
448.    Quadrates.    The  unit.s  in  tliis  case  have  been  given. 
Dedekind  Algebras.    These  have  been  considered. 
Order'  7.    ei  =  (110) 


«?5  =  (330) 
Order  8.    Type  g,  x  (>?,  i) 


e,  =  (120) 
r„  =  (l30) 


e,  =  {2\0) 
e,  =  (230) 


Xt  c, 


I'^o- 


X       Xo 


X., 


This  is  biquaternions. 

Type"  Q,  +  (>?,  i)  +  e^ 


X^  Cq  X 


2  — 


=  0 


Xi  6i 


1  ^0' 


X^ 


X       Xg 

x^  eo  —  X 


(Xfi  Cq  —  x)-  =  0 


ei=  (220) 


(110),  (120),   (210),   (220),   (330),   (331),   (l3l),  (231) 
Order  12.    Triquaternions. 

Order  16.    Quadriquaternions. 

It  is  not  a  matter  of  much  difficulty  to  work  out  many  other  cases,  but 
the  attention  of  the  writer  has  not  been  called  to  any  other  cases  which  have 
been  developed. 


ISOHEFFBRS  Q,. 


SSCHEFFERS  Qj,  Q,. 


eo 

ei 

'0 

eo 

Cl 

ei 

«! 

— fo 

PART    lir.    APPLICATIONS. 
XXni.     GEOMETRICAL. 

449.  The  chief  geometrical  applications  of  linear  associative  algebras  have 
been  in  Quiilernions,  Octonions,  Triquateniions,  and  Alternate  Numbers. 
These  will  be  sketched  here  very  briefly,  as  the  treatises  on  these  subjects  are 
very  complete  and  easily  accessible.  What  is  usually  called  vector  analysis 
may  be  found  under  these  heads.  There  are  two  other  algebras  which  find 
geometrical  application  in  a  way  which  may  be  extended  to  all  algebras. 
These  will  be  noticed  immediately.' 

450.  Eqiiipollences.    The  algebra  of  ordinary  complex  numbers 

has  been  applied  to  the  plane.  To  each  point  {x,  y)  corresponds  a  number 
2  =  a;  +  Z/*^!-  The  analytic  functions  of  z  (say  /(z)  where  df  .%■=■  f  {z) .  dz) 
represent  all  conformal  transformations  of  the  plane  ;  that  is,  if  z  traces  any 
figure  Cj  in  the  plane, /(z)  traces  a  figure  C,  such  that  every  point  of  Cj  has 
a  corresponding  point  on  C^  and  conversely,  and  every  angle  in  C^  has  an 
equal  angle  in  Co  and  conversely.^ 

451.  Equitangentials.    The  algebra 

ej 

has  also  been  applied  to  the  plane.  The  analytic  functions  of  z  represent  the 
equisegraental  transformations  of  the  plane,  such  that  /(z)  converts  a  figure 
into  a  second  figure  which  preserves  all  lengths.^  To  2=  x  +  Cj  ?/  corresponds 
the  line  ^  cos  a;  +  >;  sin  a;  —  y  =  0. 

452.  Quaternions.  Three  applications  of  Quaternions  have  been  made  to 
Geometry.  In  the  Jirst  the  vector  of  a  quaternion  is  identified  with  a  vector 
in  space.  The  quotient  or  product  of  two  such  vectors  is  a  quaternion  whose 
axis  is  at  right  angles  to  the  given  vectors.  Every  quaternion  may  be 
expressed  as  the  quotient  of  two  vectors. 

'  See  Bibliography  of  Quaternio7is.     Also  the  works  of  Ha.miltox,  Cliffokd,  Combebiac,  Gkassmanx, 
GiBBS  and  their  successors. 

'BeLLAVITIS  1-16  ;     SCHEFFERS  10.  '  SCHEFFERS  10. 

113 


e. 

ei 

<'o 

% 

ei 

ei 

ei 

0 

114  SYNOPSIS  OF  LINEAR  ASSOCIATIVE  ALGEBRA 

The  following  formulae  are  easily  found  : 

(1)  If  a  is  parallel  to  (3 F .  a/3  =  0 

(2)  If  a  is  perpendicular  to  /3 S  .  a^  =  0 

(3)  The  plane  through  the  extremity  of  ^,  and  perpendicular 

to  a  is iS{p  —  b)  a  =  0 

(4)  The  line  through  the  extremity  of  a,  parallel  to  jS F(p  — a)  jS  =  0 

(5)  Equation  of  collinearity  of  a,  /?,  7 V{a  —  /?)  {^  —  y)  =  0 

(6)  Equation  of  coplanarity  of  a,  /3,  7,  ^ S{a  —  ^)  {(3  —  y)  (7  —  ^)  =  0 

(7)  Equation  of  concyclicity  of 

a,  /?,  y,  ^ y{o^ - 1^)  (^ - r)  (y-^)  (5-a)  =  o 

(8)  Equation  of  cosphericity 

of  a,  13,  r,h,s S{a-P)  (/3-y)  [y-h)  (5  -  e)  {s-a)  =  0 

(9)  The  operator  q{)q~^  turns  the  operand  ()  through  the  angle  which  is 
twice  the  angle  of  q,  about  the  axis  of  g-.  The  operand  may  be  any  expression, 
and  thus  turns  like  a  rigid  body.  These  operators  give  the  group  of  all 
rotations.^ 

(10)  The  central  quadric  may  be  written  /5p<?)p  =  —  1  =■  gf  -\-  2  iS?.p  S^p, 
where  4)  is  a  linear  vector  self-transverse  function  ;  /I  and  fi  are  the  cyclic 
normals; 


a  =  ^gz  —  gi  i  +  *</gz—g2 /^       2^  V  =  ^9^  —  9i  *"—  '^93  —  9z^ 

i  and  7c  being  in  the  direction  of  the  greatest  and  the  least  axes,  and  the  axes 

are  given  by  </i  =     %- ,  gi  =  j^ ,  93  =  —tt-     Conjugate  diameters  are  given 

by  Sa^^  =  S^S^y  =  Sy^a  =  0. 

(11)  For  any   curve,    p  =  <^(^),    any  surface,    p  =  ^(/,  ?()    or    F{p)  =  {). 

dp   is  parallel  to  the    tangent    of   a   curve,    ^     7  //"   ^^  ^^^^  vector  curvii- 

ture,    Udp  S  y'/^ri     is  the  vector  torsion,     a  =  Udp     is  the   unit  tangent, 

^  =.  JJVdpd^p  Udp  is  a  unit  on  the  principal  normal,  y  =  UVdf  d"p  is  a  unit 
on  the  binormal.  Forasurface  i^(p)=0,  yi^  is  the  normal,  S{p—p^  v^o  =  0 
is  the  tangent  plane.* 

The  second  application^  of  quaternions  to  geometry  is  by  a  homogeneous 
method.  In  this  the  quaternion  q  is  written  q  =  Sq{l  +  p),  and  q  is  regarded 
as  the  affix  of  the  point  p  with  a  weight  Sq  =  w. 

1  Catlev  10.  'Hamilton's  works,  Tait's  works,  Joi.y's  works. 

'This  applicatioa  may  be  followed  in  Joly  30,  11,  35;    Shaw  3;    Chapman  4;    see  also  Bkij,l  1. 


GEOMETRICAL 

We  write  also 

A  .  q  Ars  -=■ 

r 

Sqr 

8 

Sqs- 

8 

t           11 

A  .  qr  Astu  =^ 

Sqs 

Sqt     Squ 

Srs 

Srt    Sru 

t 

U             V 

A  .  qrs  A  .  tuvw  = 

Sqt 
Sit 

S7/11    Sqv 
SJ-ii    Srv 

Sst 

Sm 

Siv 

115 


w 

Sqio 

Srw 

Saw 

In  pai'ticular  we  may  write 

—  A.l  Aah  =  A' .  ah 
A  .  ah  Aijh  —  A".  ab  =  V  .  Va  Vh 
S  .  a  Abe  Aijk  =S  .VaVb  Vc  =  SA  .  abc 
A  .  abc  :=  —  K .  A  .  abc  A 1  ijh 
S  .  a  A  .  bed  =  —  Sd  Abed  A  .  1  ijk 
We  have 

(1)  The  equation  of  line  a,  b  is    A  .  abq  =  0. 

(2)  The  equation  of  plane  a,  h,  g\s    S  .  q  Aabc  =^  0. 

(3)  a,  b  and  c,  d  intersect  if    S .  a  Abvd^=  0. 

(4)  The  point  of  intersection  oi  S  .lq-=-0  =■  Smq  =  Snq  is   q=-  A  .  Imn. 
The  third  application  of  quaternions  is  to  four-dimensional  space. ^ 

(1)  Any  quaternion  p  represents  a  four-dimensional  vector  in  parabolic 
space.  All  vectors  parallel  to  p,  in  the  same  sense,  and  equal  in  length  are 
represented  by  p. 

(2)  If  g^  is  a  second  vector,  then  the  angle  Z  {p,  q)  being  Q 

cos  Q  =  SUp  U^  =  S.llJiUq 

(3)  The  condition  that^  is  perpendicular  to  q  is  Spq  =■  Spq  =  0. 

(4)  There  is  for  p  as  a  multiplier  />  ()  a  system  of  invariant  planes,  one 
througli  any  given  line  q,  called  a  system  of  in-parallel  planes.  Multiplication 
by  p,  0  p,  has  also  a  system  of  invariant  planes,  called  by-iKirallel  plants,  one 
through  each  line  q.  The  displacement  of  (7  in  any  invariant  plane  is  constant 
and  equal  to  the  angle  of/).  The  tensor  of  (7  is  multiplied  by  the  tensor  oi p. 
If  3'  is  resolved  parallel  to  two  invariant  planes  of /),  these  components  turn  in 
their  planes  through  Z  p,  and  the  product  pq  has  these  results  for  its 
components. 

(5)  If  Vqp  =  0,  g-  is  parallel  to  p. 

(6)  The  projection  of  q  on  })  is    Up  Sq  KZTp. 

The  projection  of  q  on  a  vector  perpendicular  to  p  is   Yq  KUp .  Up. 

(7)  The  plane  through  the  origin  and  the  two  vectors  from  the  origin 

a,  —  (X2   and    ai  («!  —  a^)    is    aip  -\-  pa^  =  0 


1  Hatha  WAV  2,  3,  4,  5;    Stuingham  4,  5,  7. 


116  SYNOPSIS  OP  LINEAR  ASSOCIATIVE  ALGEBRA 

The  plane  through  the  point  aj  a  containing  the  vectors 

ttj  —  Uo    and    tti  (ttj  —  tto)    is    aj  g  +  quo  +  2a  =  0 

(8)  If  " 

ai  =  ±  UVec,     ao=:±  f7Fec,  and    a  =  —  aj  Oq 

then  the  equation  of  the  plane  through  Uq  containing  the  vectors  c,  e  is 

ttjjp  4-^Jao  +  2a  ^  0 

(9)  The  plane  through  c,  d,  e  is  given  by  the  same  equation  with 

«!  =  UV{cd  +  d~e-\-  ec)  a.  =  UV{dd  +  de  +  ec) 

a  ^  —  M"i  ^  "I"  ^'^2)  ^  —  i  («!  <Z  +  c^ag)  :=  —  ^  («!  e  +  eag) 

(10)  The  normal  to  the  plane  is  a^a. 

(11)  The  point  of  intersection  of  the  two  planes 

ttip  +i)a2  +  2a  =  0  =  Pii^+p^o  H-  26 
is 

/?!  a  —  a^So  +  ai  6  —  fcag 

-^  S{a2^2  —  ^i/iJi) 

(12)  If  the  two  planes  through  the  origin    (aj  ag  0)    {^i^^Q)    meet  in  a 
line  through  the  origin,  it  is  necessary  and  sufficient  that 

The  cosine  of  the  dihedral  angle  between  the  planes  is    ±  Sa^^i  =  ±  Sa..^^- 
They  are  perpendicular  when  this  vanishes. 

(13)  The  two  planes  {ayaz^a)  {^i[^o2b)  meet  in  a  straight  line  if 

ai  6  —  6  tto  +  /3i  a  —  a  /^a  =  0 
Let 

/=aiZ) — Sag         g=-^ia  —  a^2.         ''^  =  '^{(^2^3 — ai/?i) 

then  if  /=  —  S'  ^  0,  the  equation  of  this  line  is 

X  —  2  Vab 

t=—f— 

(14)  The  two  planes  meet  in  a  point  at  infinity  if  wj  =  0  and  /+  </  ij:  0  ; 
they  meet  in  a  line  at  infinity  if 

/6?i  =  ±  ai  /i?2  =  ±  ao 

(15)  The  perpendicular  distance  between  the  planes  (aia3  2a)  (aj  a,  26) 
is  in  magnitude  and  direction  aj  (a  —  6). 

(16)  The  vector  normal  from  the  extremity  of  c  to  the  plane  (aiajZi)  is 

^  ttj  ( 2a  +  «i  c  +  Ctto) 

(17)  The  vector  normal  from  the  origin  to  the  intersection  of 

oh  —  ha  .  ,  , 

(aia2  2a)  and  (/3]^2  26)is y  +  r/  =  0,    /4:gr 

OOC1  —  fXo  0 

(18)  Two  planes  meet  in  general  in  a  point  or  in  a  straight  line.    Through 
any  common  point  transversal  planes  may  be  passed  meeting  the  two  in  two 


GKOMBTRICAL  1  1  7 

straight  lines  m,  v  and  forming  with  them  equal  opposite  interior  dihedral 
angles.  The  angle  between  these  lines  u,  v  is  the  isoclinal  angle  of  the  two 
planes.  Two  planes  have  maximal  and  minimal  isoclinal  angles  if  there  exist 
solutions    c-\-u\ti   and    c  +  (Sfv   of  their  equations  such  that 

)Sai uv  =  0         S^i MW  =  0         Suv  df:  Sa^ u^^v 
The  planes  of  these  angles  and  these  only  cut  the  given  planes  orthogonally. 
The  lines  u  and  v  are  given  by 

«  =  «!  (yi  +  nO  —  {yi  +  72)  «3  yi  =  UVrxi  (3 

v  =  ^i  iyi  +  r,)  -  (/I  +  y-z)  (^2  n  =  ^^«3  /?i 

«i "  =  «'  =  «i  (ri  —  r^)  —  (/i  ~  /z)  «2 

^iV  =  v'  =  (3i{yi  —  /»)  —  iyi  —  /a)  (3^ 

(19)  There  are  no  maximal  and  minimal  isoclinal  angles  if  any  one  of 
the  four  conditions  is  satisfied  : 

/?!  =  ±  ttj  /So  =  ±  a2 

In  this  case  the  isoclinal  angle  is  constant  for  all  variations  of  0. 

(20)  Two  planes  are  perpendicular  and  meet  in  a  point  if 

Sai  /?i  =  —  Sa^^z  4:  0  or  :}:  1 
Two  planes  are  perpendicular  and  meet  in  a  line  if 

Two  planes  are  hyperpendicular  if  every  line  in  one  is  perpendicular  to  every 
line  in  the  other.     In  this  case 

^ai  /3i  =  —  /Sixg  /?2  =  ±  1 
that  is 

/?j  :=  ±  tti  /?o  =:  ^  a.. 

If  two  planes  are  parallel 

Ui  =  /?i  ag  ^  /?3 

453.    Octonions.    The  following  are  the  simpler  results : 

(1)  The  vector  from  0  to  P  is  a  rotor  p  and  may  be  transferred  anywhere 
along  its  own  line.  It  is  not  equal  to  any  parallel  rotor.  Rotors  from  the 
same  point  O  are  added  like  vectors,  p  +  e  being  the  diagonal  of  the  parallelo- 
gram whose  sides  /7wn  0  are  p  and  e. 

(2)  The  side  parallel  to  p  is   p  -f  flil/ep,  that  parallel  to  e  is   s  +  HJ/pe. 

(3)  If  all  vectors  are  drawn  from  0,  the  usual  formulae  of  quaternions 
hold.  Thus  the  equation  of  the  plane  perpendicular  to  6  through  its  extremity 
is  S  {p  —  (S)  f^  =  0 ;  the  line  through  the  extremity  of  8  parallel  to  a  is 
p  :=  h  -\-  ta.  But  a  rotor  in  tlie  plane  is  not  p  —  6  but  p  —  b  -\-  D.M .  hp  and 
a  rotor  on  the  line  is  not   xa    but    x  (a  +  HJ/rW). 

(4)  A  velocity  of  rotation  about  an  axis  is  represented  by  a  rotor  on 
that  axis,  a  translation  along  the  axis  is  a  lator  on  that  axis.  A  motor,  as 
G)  -\-  HcT,  indicates  a  displacement  such  that  in  time  dt  any  point  rotates  about 
the  axis  of  the  motor  by  an  angle  71) .  dt  and  is  translated  along  the  axis  by 
a  distance  Tadt. 

8 


118  SYNOPSIS  OF  LINEAR  ASSOCIATIVE  ALGEBRA 

(5)  The  axis  of  M .  AB  is  the  common  perpendicular  of  the  axes  of  A 
and  B.  The  rotor  of  J/.  AB  is  the  vector  of  the  product  of  the  rotors  of  A 
and  B  considered  as  vectors  through  0.  The  lator  of  M.  AB  has  a  pitch  equal 
to  the  sum  of  the  pitches  of  A  and  B  and  the  length  of  the  common  perpen- 
dicular multiplied  b}'  the  cotangent  of  the  angle  between  A  and  B  (=  d  cot  0). 

(6)  The  rotor  of  J.  -f  -B  is  equal  and  parallel  to  aj  -|-  /3,  the  sum  of  two 
rotors  from  0  equal  to  the  rotor  of  A  and  parallel  to  the  rotor  of  B  respect- 
ively, and  intersects  the  common  perpendicular  from  J.  to  i?  at  a  distance 
from  0  equal  to  [w  being  the  common  perpendicular] 

T  \m  SI3  {a,  +  13)-'  +  {p~  p')  Ma,  /?.(«!  +  /?)-==] 

(7)  S, .  ABC  is  one-sixth  of  the  volume  of  the  parallelopiped  whose  edges 
are  the  rotors  of  ABC.  M, .  ABC  is  a  rotor  determined  from  the  rotors  of 
A,  B,  C  &s  V .  a^y  is  from  a,  /?,  y. 

t .  S .  ABC  =  tA->rtB-\-tG+  dcotd  —  etan^);  d  and  0  as  in  (5),  e  and  ^ 
the  common  perpendicular  from  M .  AB  to  C,  and  the  angle. 

,_,„„         A    ,    .r,   ,    .^  dcotd  —  etand) 

t.MABG=tA+  tB  +  tC  —  ^Towi — o—-; ,-a  ,+ — ^ 

cot-t/tan-^ -f  cot-y +  tan-^ 

(8)  If  5  and  C  are  motors  whose  rotors  are  not  zero  and  not  parallel, 
then  ^B  -\-  YG  is  any  motor  which  intersects  the  common  perpendicular  of 
B  and  C  perpendicularly.^ 

454.  Triquaternions.  If  /<,  ^i'  are  points,  h,  b'  lines,  w  and  ro'  planes,  all  of 
unit  tensor, 

{i  =  ^j-o  +  (0  {ixi  +  jxo  +  Z'a-g)    [I   is  the  point  -^- ,    ^ ,    -f 

a  =  (j/?„  +  ^  (iaj  +  joi2  +  /I'ttg)    a  is  the  plane  /?o  ^o  +  «i  ^i  +  «2  ^^s  +  ^3  a^s  =  0 
b  =iai+  jao  +  kas  +  o  {ii^i  +  j^.^  +  ^^^s)     «i  Pi  +  <^z  P-z  +  "s  /^3  =  0 

h  is  the  line    1^  .,   P^  .,    Pl^^  ,  1^  ,  J;^.  .,    P^ 
tti  ttg  as         /:Ji         /^3         /:^3 

That  is,  a  point  or  a  plane  is  represented  by  the  symmetry  transformation 
it  produces;  a  line,  by  a  rotation  about  it  as  an  axis  through  180°. 

(1)  Ghh'    is  —  cos  of  angle  between  the  lines. 

(2)  Gmm'  is  —  cos  of  angle  between  the  planes. 

(3)  L(i^'   is  the  vector  of  n'  towards   {x. 

(4)  L^h   is  the  vector  perpendicular  of  the  plane  containing  the  point  and 

the  line,  tensor  equal  to  distance  from  point  to  line. 

(5)  Lhh'   is  the  complex  whose  axis  is  the  common  perpendicular  and 

whose  automoment  is  tlie  product  of  the  shortest  distance  by  the 
cotangent  of  the  angle. 

(6)  Liim  is  the  perpendicular  drawn  through  ^  to  the  plane  m. 

(7)  Lhm  is  the  point  of  intersection  of  the  line  and  the  plane,  tensor  equal 

to  the  sine  of  the  angle  of  the  line  and  plane. 

'M'AULAT  2. 


GEOMETRICAL  1  ]  9 

(8)  Lmm'  is  the  line  of  intersection  of  the  two  planes,  tensor  equal  to  the 

sine  of  the  angle. 

(9)  P^ih  is  the  plane  through  ^  perpendicular  to  h. 

(10)  Phh'  is  the  plane  at  oo  multiplied  by  the  shortest  distance  and  the  sine 

of  the  angle  of  the  two  lines.     7hh'  is  the  moment  of  the  two  lines. 

(11)  P^m  is  the  plane  at  ro  multiplied  by  the  distance  from  fi  to  m  and 

positive  or  negative  as  ^  is  on  the  side  of  the  positive  or  negative 
aspect  of  the  plane. 

(12)  P8m  is  the  plane  drawn  through  the  line  perpendicular  to  the  plane, 

tensor  equal  to  the  sine  of  the  angle  of  the  line  and  the  positive 
normal  of  the  plane. 

(13)  If  y,  y'  are  two  complexes  of  unit  tensors,  Vyy'  =:  0  means  the  two 

are  in  involution. 

(14)  A  displacement  without  deformation  is  given  by  r()r~^: 

r  =  q  +  aqi  Sqqi  =  0  P  [r' —  (Xr)^]  =  0 

The  axis  is  5  =  VLr  =  U{Vq  +  a Vq^). 

a 

The  angle  of  rotation  is  2d.         6  =  tan~^   tv~ 
The  translation  is  2yj.         ri 


_  Sq, 


TVq 
r  =  (1  +  cjrV)  (cos  0  +  5  sin  6) 

(15)  Transformations  by  similitude  are  given  by  j- =  fiq  -\-  aq^.      Sqqi  =  0 

(16)  The  triquaternion  ?•  produces  a  point  transformation  m'  =  rmr~^, 

if  r  =  ?u  +  7  +  p,  2wp  ~.FP=0 

This  transformation  may  be  written  ^ -,  which  is  a 

rotation  about  the  line  d,  and  a  homothetic  transformation  whose 

center  is  OT  and  coefficient        i^  rr    • 

w  -\-  Im 

Hence  r  produces  the  group  of  transformations  by  similitude.' 

(17)  A  sphere^  is  represented  by  the  inversion  which  it  leaves  invariant; 
that  is,  by  the  quadriquaternion    ^  (z'xj  -^-jiji  +  Icz^  +  uirj  +  u'w^. 

(18)  If  J/ and  M'  are  two  spheres  of  zero  radius,  m  and  m'  their  centers, 
L  m  M' ■=■  L  m' M  is  the  line  {mm').  The  sphere  on  w?  m'  as  diameter  is 
Pm  M'.     If  cZ  is  a  line,  then  P  .  Md  is  the  plane  through  d  and  m. 

455.  Alternates.  Tliere  are  various  applications  of  the  different  systems 
of  alternates,  notably  those  which  are  called  space-analysis — the  development 
of  Grassmann's  systems;  vector-analysis — a  Grassmann  system  without  the 
use  of  point-symbols  or  else  a  system  due  to  GiBBS;  and  finally  the  Clifford 
systems.     No  brief  account  or  exhibition  of  formulae  can  be  given.^ 

'COMBEBIACa.  'COMBBBIAO  3. 

'See  Bibliography  of  Quaternions;    notably  Jolt  6 ;    Htde  4  ;    Whitehead!;    Gibbs-Wilson  3. 


120  SYNOPSIS  OF  LINEAR  ASSOCIATIVE  ALGEBRA 

XXIV.     PHYSICO-MECHANICAL  APPLICATIONS. 

456.  These  are  so  numerous  that  they  may  be  only  ghmced  at.  Quater- 
nions has  been  applied  to  all  branches  of  mechanics  and  physics,  biquaternions 
and  triquaternions  to  certain  parts  of  mechanics  and  physics,  alternates  and 
vector  analysis  in  general  to  mechanics  and  physics.  The  standard  treatises 
already  mentioned  may  be  consulted. 

XXV.     TRANSFORMATION  GROUPS.^ 

457.  Theorem.  To  every  linear  associative  algebra  containing  a  modulus 
belongs  a  simply  transitive  group  of  linear  homogeneous  transformations,  in 
whose  finite  equations  the  parameters  appear  linearly  and  homogeneously,  and 
conversely." 

458.  Theorem.  Associated  with  every  linear  associative  algebra  containing 
a  modulus  and  of  order  r,  is  a  pair  of  reciprocal  simply  transitive  linear 
homogeneous  groups  in  r  variables.^ 

459.  Theorem.  To  a  simply  transitive  bilinear  group  which  has  the 
equations 

XJ=  2  o.M.Xu  -^—  (t  =  1  .  .  . .  r) 

s 

corresponds  the  algebra  whose    multiplication    table  is   €^6/,=:  "S.  a,^,  e^,  and 

s 

conversely.* 

r  r 

460.  Theorem.    The  product  of  a  =  2  aiCj  and  h  =.1,h^ei  gives  the  finite 

transformation  corresponding  to  the  successive  transformations^  of  the  para- 
meters («!....  Or)  and  (Z>i  ....  h^). 

461.  Theorem.  To  every  sub-group  of  G,  the  group  corresponding  to  the 
algebra  2,  corresponds  a  sub-algebra  of  2,  and  conversely.  To  every  invariant 
sub-algebra  of  2  corresponds  an  invariant  sub-group  *  of  G. 

462.  Theorem.  To  the  nilpotent  sub-algebra  of  2  corresponds  a  sub-group 
of  (r,  Fi  (/)... .  Yk  (/),  such  that  for  no  values  of  Y ./or  X ./,  transforma- 
tions respectively  of  the  sub-group  and  the  group,  do  we  have** 

Y{X/)  =  c,Xf  c.  to 

X{  Yf)  =  JXf  <y  1 0 

( FX)  =  Y{Xf)  -  X{  Yf)  =  <.A7  c.  t  0 

463.  Theorem.  The  invariant  sub-group  g,  corresponding  to  the  nilpotent 
sub-algebra  a,  is  of  rank  zero.* 

'Stddt7.  'Poinoare  1,  2,  3;   Study  8;    Cautan  3.     See  also  Schur  1. 

3Studt1,3;  Lie-Sciieffeks  4;  Cautan  3.  *Cartan  3.  'Cautan  3;  Study  3. 

'Cautan  3.  Cf.  Engel,   Kleincic  Beitriige  zur  Gruppentheorie,  Lcipziger  Berichte,  1887,  S.  'J6;    18',)3, 
8.  360-369. 


TRANSFORMATION  GROUPS  121 

464.  Theorem.  To  every  quadrate  of  order  r  ■=■]>"  corresponds  the  para- 
meter group  of  tlie  linear  homogeneous  group  o\' £)  variables.* 

465.  Theorem.  To  every  Scheffers  or  Peiuce  algebra  corresponds  an 
integrable  simply  transitive  bilinear  group,  whose  infinitesimal  transforma- 
tions are 

o  -a 

Xi  =  Xi  23-  +  -  2/p  3y  *  =  /3p    («( ,  ^i  are  the  characters  of  r^i) 


and  whose  finite  equations  are^ 

2/1  =  <%  Vi  +  ^i  a-..  +  2  a,^,  h^  y,  (>.  <  i,  /t/  <  i j 


A,  (* 


466.    Theorem.    Every   simply  transitive  group  can   be    deduced    from    a 
group  of  the  form  just  given, 

y»  =  ^'"i^-gfar  +  2  a,,,  r^^  -g^(.r    («>  ^  s>J,  A=«o  «»=«i,  /3.=/?0 

or 

by  setting  to  correspond  to  each  variable  A''*"'  or  F'*"^  of  character  (a/3),  p^P^ 
new  variables  a'^',  y'-^\  where  i,j  are  respectively  any  two  numbers  of  the 
series  1,  2  ....  p„,  1,  2  ....  ^)^.  Likewise  to  each  parameter  J."'',  JS'"'  of 
character  (a/?),  Palh  new  parameters  a|j',  h'^\ 

The  simply  transitive  group  is  then  defined  by  the  infinitesimal  trans- 
formations 

X%  =  2  xt  -^  +"  2"'>i  -g4-         (/?P  =  i;  a,  ^  =  1,  L>  . . . .  i..) 
Pi  g  8 

or  by  the  finite  equations" 

1,2. ...Pi 

A 

«/(i)   V  „(^.)    ,,,(t)      I      5"   ft(i)    -(<■.)    _|_    y  „  ^C")    ,,(f) 

2/  a?  —  "A3     yaA    T^   -i   ^\^   ""^aX      T^    ^  CCp^j    O^p     ^^ 

A  A  fwA 

'CARTAN2;  MoLiEN  1.     Cf .  Cayle Y  11,  5 ;    Laouerre  1 ;   Stephanos  1  ;   Kleis  1 ;  Lipschitz  2.    Also 
Cayley  3  ;    Frobenius  1;    Sylvester  1  ;    Weyr  5,  6,  7,  8. 
«  Cartan  3. 


122  SYNOPSIS  OF  LINEAK  ASSOCIATIVE  AX.GEBRA 

467.  Theorem.  Every  simply  transitive  bilinear  group  G  is  formed  of  a 
sub-group  r  of  rank  zero^  and  a  sub-group  g  which  is  composed  of  h  groups 
(Ji  •  ■  •  •  9hf  respectively  isomorphic  with  general  linear  homogeneous  groups  of 
Ih)  Pz  •  ■  •  •  Ph  variables.  Moreover  the  variables  may  be  so  chosen  that  the 
p\  first  variables  are  interchanged  by  the  first  gi  of  these  h  groups,  like  the 
parameters  of  the  general  linear  homogeneous  group  on  p^  variables,  and  are 
not  altered  by  the  other  h  —  1  groups ;  the  same  is  true  of  the  p\  •  ■  ■  ■  p\ 
following  variables ;  finally  these  p\+  .  ■  •  ■  -^  p\  variables  are  not  changed 
by  the  sub-group  ^  V. 

468.  Theorem.  All  simply  transitive  groups  are  known  when  those  in 
§  465  are  known." 

469.  Theorem.  Every  real  simply  transitive  group  G  is  composed  of  an 
invariant  sub-group  V  of  rank  zero,  and  a  sub-group  g  which  is  the  sum  of  h 
groups  gi  ■  ■  ■  ■  Qh,  each  of  which,  belongs  to  one  or  other  of  the  three  types 
following: 

(1)  The  groups  of  the  first  type  are  on  p?  variables  xy  and  are  given  by 
the  formulae 

^V  -  ^li  ^y.^,  -+-  ^zi  g^^^  -1-  •  •  •  •  +  a-pf  2^^^ 
or 

•*•  ij  -~  ^U  •'^il  T  ^2j  "^12  "T    •  •  •  •      I     (tpj  ^ip 

giving  the  parameter  group  of  the  general  linear  homogeneous  group  on  p 
variables. 

(2)  The  groups  of  the  second  type  are  on  the  2p^  variables  Xij,  y^  and  are 
given  by  the  formulae 

^ij - o^u  Q^^,+  ■■■■  +  Xpi a^.  +  2/ii a^  +  •  •  •  •  +  Vpi dy^^ 


^--^"3^,,+  ••••  +  ^''^  3^,,  -  2^1^  Bx,,  +  ••••-y-*ax,, 


or 


X  ij  —  (iij  Xfi  -\-  ■  ■  ■  ■  +  cij,j  ^ij,  —  6]j  ?/ii        ....  —  bpj  yip 

y'ij  =  «];  2/ii  + +  Cpj  !/ip  +  h  a^ii  + +  hj  ^iP 

(3)  Those  of  the  third  type  are  on  4p^  variables   Xy,  i/ij,  2,j,  t^j,  given  by 
the  formulae 


\j         <^y\j         ^*Aj         WAj 

3  3  ..      3      ,   .      3 

<^!/\j  ^'^/^j  '"-A^-  ^^>~j 

3  3,3 


Y,j-^^^(x,,  3,„  -2/A.  a^--^^i:a^  +  ^^*  dzJ 


y  _i (      3,        a  a       ,3\ 

^'^  -  i  K""''  3^  +  ^«  -3^:7  -  '^*  3^  ~  ^''  3^,, ; 

„,         '.'  /        3  3      ,  3         ,      3    \ 


iCartan3.     Cf.  Molien  1.  «Cabtan: 


TRANSFOUMATION  GROUPS  1  23 


or  p 

x'y  =  i;  {a^j  JCu  —  ^Aj  y.A  —  fh}  Z(A  —  f^A^  <<a) 

A  =  l 
V 

y'ij  =  1  (a, J  yt^  +  h^^  Xo,  —  c^^  ti^  —  d^j  z,J 

A  =  l 
V 

A=l 
P 

fij  =  2  (a,,-  ti,  —  b,j  Zi^  +  6\;  ?/„+  (Z,,-  a-(,) 

A  =  l 

To  each  of  these  groups  o£p^,  2p^,  or  4^/  variables  we  can  set  to  corre- 
spond p^,  2p^,  or  4^/  variables  which  are  interchanged  by  these  equations, 
without  being  changed  by  the  other  groups  which  enter  g  nor  by  the  sub- 
group r.     All  these  variables  are  independent.^ 

470.  Theorem.  The  groups  in  §469  are  not  simple,  but  are  composed  of 
an  invariant  sub-group  of  one  parameter  and  simple  invariant  sub-groups  of 
p^ —  1,  2p"  —  1,  4jr —  1  parameters.^ 

471.  Theorem.  Simply  transitive  bilinear  groups  in  involution  (transfor- 
mations commutative)  are  given  by  the  formulte 

(1)      ^=x3^-f22/./-  ^'="4. +5"^"^^'!: 

i=l,  2 r—1  oL^is  =  0    if  s^i,  8^2. 

or 

x'  =  ax  y'i  =  a!/L  +  hiX+  1  a,^;  b^  y^ 


Kv- 


3x 


^  P\  0\  o 


or^ 


a;'  =.  ax  —  cz 


z'  ■=.az  -\-  ex 

y'i  =  ay^—  ct^  +  liX ~  diZ  -\- 1  a,^i  (6^  y,  —  d^  Q  —  1 13,^^ {b^  t,  +  d^  y,) 
t'i  =  at,  +  cy^  -f  b,  z  +  d,  .t  +  S  a,,^  {b,  t,  -f  d,  y,)  -f  2  /?.,.■  [b,  y,  -  d,  Q 

472.  Theorem.  Every  bilinear  group  G  is  composed  of  an  invariant  sub- 
group r  of  rank  zero,  and  of  a  sub-group  g  which  is  the  sum  of  a  certain 
number  h  of  groups  which  are  respectively  isomorphic  with  the  general  linear 

•  Cabtan  2. 


124  SYNOPSIS  OF  LINEAR  ASSOCIATIVE  ALGEBRA 

homogeneous  group  onp^,  p^  ■  ■  ■  •  Th  variables.  Every  real  bilinear  group  G 
is  composed  of  a  real  invariant  sub-group  V,  and  a  sub-group  g  which  is  the 
sum  of  h  groups  each  isomorphic  with  one  of  the  three  following  groups: 

(1)  The  general  linear  homogeneous  group  on  ^  variables. 

(2)  The  group  on  2jf  parameters  and  2p  variables  a:,-,  yi'. 

x\  =  ciii  xi  + +  Opi  a-p  —  &i;J/i  — —  ipi  l/p 

2/i  =  «i(2/i  + +  «pi  Z/p  +  ^li  a-i  + +  bpi  Xp 

(3)  The  group  on  4p^  parameters  and  4p  variables  a-^,  ?/;,  Zj,  i^: 

p 
x'i  =  2  (a^i  x^  —  i,i  y^  —  c^(  z^  —  d^i  t^) 

P 

y\  —  2  {a>,^  y^  +  ft^^  x^  +  c^^  t^  +  d^  zj 

A=l 

V 

z'i  =  S  [a^i  z^  +  Z>^j  t^  +  c^i  x^-\-  d^i  y^) 

A=l 

V 

t\  =  2  (a^i  t^  +  6^i  z^  +  c„.  ?/^-f  rf;^,  xj 

Each  of  these  groups  is  formed  of  a  simple  invariant  sub-group  on  p^ — 1, 
2p~ —  1,  or  Ap? —  1  parameters  and  an  invariant  sub-group  on  one  parameter.^ 

473.  Theorem.  Every  bilinear  group  G  is  composed  of  an  invariant  sub- 
group r  of  rank  zero,  and  one  or  more  groups  g^,  Qn  .  .  ■  .  of  which  each  g  is, 
symbolically,  the  general  linear  homogeneous  group  of  a  certain  number  of 
variables  X^  .  .  .  .  X^,  these  variables  being  real,  imaginary,  or  quaternions, 
and  the  p^  parameters  having  the  same  nature, 

X'(p=2X«A. 

A  =  l 

If  the  variables  and  the  parameters  of  the  bilinear  group  G  are  any 
imaginary  quantities  whatever,  the  group  is  composed  of  an  invariant  sub- 
group r,  of  rank  zero,  and  of  one  or  more  sub-groups  g^,  g^  ■  ■  ■  ■  of  which 
each  g  is  the  general  linear  homogeneous  group  of  a  certain  number  of  series 
of  p  variables,  of  course  imaginary.^ 

474.  Theorem.  The  quaternion  algebra  is  isomorphic  with  the  group  of 
rotations  about  a  fixed  j^oint,^  with  the  group  of  projective  transformations  on 
a  line,  and  with  the  group  of  special  linear  transformations  around  a  point  in 
a  plane. 

475.  Theorem.  Biquaternions  is  isomorphic  with  the  group  of  displace- 
ments in  space  without  deformation.^ 

476.  Theorem.  Triquaternions  is  isomorphic  with  the  group  of  displace- 
ments and  transformations  by  similitude.*  Quadriquaternions  is  isomorphic 
with  the  group  of  conformal  transformations  of  space. 


'  OAETAN  2.  'CAYI.EY  10;     I.AOUKKllE  1  ;     STEPHANOS  1  ;    SxitlNdllAM  S;     BkezI. 

'M'AULAT  2;    COMBEBHC  I  ;    Stl'dt  5.  *  Combebiac  2,  3. 


AnSTIlACT  GROUPS  |  25 

XXVI.     ABSTRACT   GROUPS. 

477.  Theorem.  Every  abstract  group  is  isomorphic  with  a  Fkobenius 
algebra  of  the  same  order  as  the  group.' 

478.  Theorem.  Tiie  expressions  for  the  numbers  of  tlie  Frobenius  algebra 
corresponding  to  the  group  are  determined  by  finding  the  sub-algebra  consist- 
ing of  all  nMmi)ers  commutative  with  every  number  of  the  algebra,  then 
determining  by  linear  expressions  the  partial  moduli  of  the  separate  quadrates 
of  the  algebra,  and  then  multiplying  on  the  right  and  on  the  left  by  these 
partial  moduli.  Every  number  is  thus  separated  into  the  parts  that  belong  to 
the  different  quadrates.  The  parts  for  any  quadrate  of  order  r^  determine 
the   rf  quadrate    units  of  the  sub-algebra   consisting  of  the  quadrate,  which 

p 
determination  is  not  unique.     In  terms  of  these  r  =  t  ij  units  all  numbers  of 

i  =  l 

the  algebra  may  be  expressed." 

479.  Theorem.  The  characteristic  equation  of  a  Frobenius  algebra  con- 
sisting of  2'  quadrates  is  the  product  of  ;j  irreducible  determinant  factors.  The 
prelatent  equation  and  the  post-latent  equation  are  identical  and  consist  of  the 
products  of  these  p  irreducible  factors  each  to  a  power  r^  equal  to  its  order.^ 

480.  Theorem.  The  linear  factors  of  a  Frobenius  algebra  correspond  to 
numbers  which  are  commutative  with  all  numbers.  The  number  of  linear 
factors  is  the  order  of  the  quotient-group;  that  is,  the  order  r  divided  by  the 
order  of  the  commutator  sub-group. 

481.  Theorem.  The  single  unit  in  each  of  the  quadrates  of  order  unity, 
may  be  found  as  one  of  the  solutions,  cr,  of  the  equations 

^(y  =  a^=ta     for  alU"'s 

For  the  ^'s  it  is  sufficient  to  take  the  r  numbers  corresponding  to  the  operators 
of  the  group.  Thus  if  cr  =  2  .  a-^  C;,  and  if  e,  Cj  =  e^j,  hence  <=•,•  e,-i^  =  e,,  we 
must  have 

ta  -=0  .  ej  =  I,  .  Xi  Cij  =  2  x^  - 1  e^      for  all  y  s 
Hence 

tXi  =  Xij~l 

If  ej  is  of  order    (ij,    ep  =  fi  =  1,    then 

Xij-i  =  tXi         Xij-2  =  f  Xi  ....  Xfj-a  =  t'Xi         fj  =  1,  or  a:,  =  0 


Hence 


27tn  ,     , — -    .    27tn 
t  =  cos \-V—l  sin =  L  n  =  1  .  .  .  .  w, 


since  not  all  a-j  vanish. 


»Poincark4;   Shaw  6.     This  theorem  follows  at  once  from  Cartan  2.     See  also  §121. 
>Poincare4;   Shaw  6.  3  Frobenius  14  ;   Suaw  6. 


1  26  SYNOPSIS  OF  LINEAR  ASSOCIATIVE  ALGEBRA 

a  =  -Exi{ei  +  tj  eij-1  +  {^Cij-z  +  .  .  .  . ) 
=  ^Xie^{l  +  (jej-i+  ....  +ip-'ej)  {j=l....r) 

The  subscripts  i  run  through  those  values  only  which  are  given  by  the 
table  \  G\  =  {eiCj].  By  operating  on  <y  with  other  numbers  e^^  we  establish 
other  equalities  among  the  a;'s  and  finally  arrive  at  the  units  in  question. 

482.  Theorem.  The  units  in  the  quadrates  of  order  2^  may  be  found  as  the 
solutions  of  the  equations 

^~<y  =  ti^a  +  f^o         o'Q  =:  <i cr^  +  <3 (7  (^  any  number) 

We  may  state  this  also 

(?i  ?2  +  ^2  ^i)  o'  —  ^  ^1  (7  —  ^2 ^2  cr  +  /3<^  =  0       (^1,  ^2  aij  numbers) 
The  units  in  the  quadrates  of  higher  orders  may  be  found  by  similar 
equations. 

483.  Theorem.  The  numbers  e^,  i  =  1  . . . .  ?•,  may  be  arranged  in  con- 
jugate classes,  the  sum  of  all  of  those  in  any  class  being  commutative  with 
all  numbers  of  the  algebi'a.     If  these  sums  are  K^,  K^  . . . .  K,^^  then 

n 

i=l 

The  partial  moduli  of  quadrates  of  order  1",  are  formed  by  operating  on  a 
with  all  numbers  and  determining  the  coefficients  to  satisfy  the  equations 

^(T  :=ta        cr^  =  ta 

The  partial  moduli  of  quadrates  of  order  2^  and  higher  orders  satisfy  the 
equations  of  §48  2. 

484.  Theorem.  Every  Abelian  group  of  order  «  defines  the  Feobenius 
algebra  ^ 

Cf  =  ;iiio  (i  =  1 r) 

485.  Theorem.    The  dihedron  groups,  generated  by  gj,  63, 

6j™  =  l=e|  €361  =  61™-!  63 

define  Frobenius  algebras  as  follows : 

When  m  is  odd :  Let  (j™  =  1,  u  being  a  primitive  root  of  unity,  then  the 
algebra  is  given  by 

/"-no     ■^220     '^2i  — 1>  2i-l>  0      '^^2i  — 1>  2i>  0      ■^2£>2i  — 1>0     ^^21 1  3(  >  0         ('  —   A....  ^  ) 

We  notice  that 

Cj  ^  ?.j]o  +  ?-320  +   ^  (<■*     *  '^i.'i  +  1 )  2J  +  1  >  0  +  '■'    '^21  +2>  2i  +  2)  o)        (    i  =   1    ....  ^        J 

Cg  =  /IjiQ  /I220  +  2  (^21  +  1  >  2i  +  2  >  0  "i"   ^^21  +  2  >  2i  +  1  >  0)  ^t  =   1    ....  -        J 

'  Shaw  0.     This  reference  applies  to  Uie  following  sections. 


ABSTRACT  GROUPS  127 

Wlien  m  i.s  even,  the  algebra  i.s  yiven  hy 

^WQ      ^"Z'iO      ^Wa      ^'<40      '^2(-l}  21- 1)  0      '^■2i—\}'il}0     ''•aO2i-l>0      '^-ZltZiid 

We  notice  tlual ' 

^1  ^   '^110  +  '^XiO  ^zm  /'.410  +  i  <■>        ~       '''•2l-l>  2(-l>  0  ^ 

+  :i-"-^'>.2„2„0      (i=3....^-+l) 

*2  — ■  '^iio  —  '''I'l.'o  +  '^yao  —  '^MO  +  "  ('^lu-ij  21. >  oH~  '^2i;  21-  J  j  0) 

486.  Theorem.    The   rotation   groups,    not   dihedron    groups,    defiiie    the 
algebras  given  below  : 

(a)  The    tetraliedral  group:  generators  Cj,  e^  e?  =  1  =  cf  =  (e,  Cg)' 
Let  u"  =  1.     The  Fkobenius  algebra  is 

'^IIO      ^^-220      '*'.Tt(l      ^UO      '^-iSA      ''-IfiO      '''510      '"-650      '''SCO      \iO      ^50      ^va 

^1  =  '^IIO  +  '■■>"  \'20  +  <■■>  "^^m  +  '^410  +  '■■>"  ''-650  +  <■■>  ?'CfiO 

^2  =  '^llO  +  ^^220  +  '^SiliO J  ('^^O  +  '^-IM  +  '^(Joo)  +  I  (?-4Bo  +  /l,oo  +  ''-CW 

+  ''-5«0  +  ^MO  H"  ''■660) 

(b)  The    octahedral    group :    e|  =  1  =  oil  ^  (e^  e^)'- 
Let  w*  =  1.     The  algebra  is 

'^UO      ^^220      ^^330     ^^440      '''•340      ''-430      '^650      ^-fifiO      ''-770     '^■SCO      '^660      '^670 
^^760      ^^070      '^760      '^880      ''■990      ''"aaO     '^890      '''•980      '^8a0      ''-a«0     ''-9a0      '^a90 

^1  =  ''"IIO  '^220  +  '''•330  ''^440  +  ''^550  +  "'  '^CBO  +  <J  '^770 '■■^  ^m  +  ^^'^  '^990  +  '■'  '^aaO 

e,  =  ;i„o  +  ^220—  i  ^330  +  i^^-  3  X430  +  M 1  +  <^)  ^sco  +  i  (1  -  w)  ^5,0 

+   i  (1  -  W)  ^890  +  i  (1   +  l.i)  Xg^o  +   Jn^—  3  ;i3,o—  i  ;*-440 

+  i  (1   +  O)  ;inBO  —   ^  ^060  +   ^  ^CTG  +  Ml  —  '■>)  ^m  +   ^  "  ?^980  +  *  ^.0 

+  i  (1  -  <,,)  X-,0  +  h  ;i:,-,o  +  i  (,)  a,:o  +  Ul  +  '^)  A^^,  +  U<.90  -  i  <^  >.„ao 

(c)  The  icosahedral  group :    ej  =  1  =  e|  =  (Cg  Cj)-. 
The  algebra  is   X„o    ^00    '^'.'o    ?-,„;o    ^sw    where - 

*,y=  2,  3,  4        7^,  Z=  5,  6,  7        j>,  <?  =  8,  9,  a,  /?        *,  ^  =  >-,  .\  e,  f,  >? 

487.  Theorem.    The  group  G^^^,   e\=\=el  =  (e.c,)",    defines  the  algebra 

''"llO     ^ijO     ^^klQ      '"ji./O      ^slO      ^uvG 

where ' 

h3  =  2,  3,  4  Ic,  1=  5,  6,  7  ^j,  y  =  8,  9,  a,  ^,  y,  8 

s,  t  =  e,^,  r„  6,  I,  X,  X  ti,v  =  ^,  r,  o,  n,  p,  a,  r,  ^ 

488.  Theorem.    The  groups  defined  by  the  relations  e"=  1  =eS,  e^'  ^  e^=^e'^ 
m  prime  to  a,  give  Fkobenius  algebras  of  order  r  =  ac  which  are  sums  of 
quadrates  as  follows  : 

ajt-g     of  order  1  Itjkj     of  order  (7; 


'Shaw  6.  «Fbobenids3;    Dickson  4.  »Poincare4. 


128  SYNOPSIS  OF  LINEAR  ASSOCIATIVr:  ALGEBRA 

where  a^  is  the  highest  cominon  factor  of  m  —  1  and  a;  g  is,  the  lowest  expo- 
nent for  which  m'-'  =  1  (mod  a);   c=-  kg. 

If  a'  is  the  smallest  divisor  of  a,  4^  1 ,  and  az^a^  a',  then  m"'  ^  1  (mod  a^). 

If  a"  is  the  next  smallest  divisor  of  a,  a  =  a^a",  mP-=.  1  (mod  a.^,  and  so 
on  for  all  divisors  of  a ;    if  also  4>  {N)  is  the  totient  of  {N),  then 

^{a)  =  hg     ^  (a,)  =  /ii(7i  ••••  ^{aj)=^h9j      {j  =  I,  2.  .  .  .  I— l,i -\- I .  .  .  .p) 

We  notice  that  if  gj  is  a  primitive  a-th  root  of  unity,  n  a  primitive  c-th 

root  of  unity 

e,  =  l  cA' '"'  ?^^- »  eo  =  2  7t*  (2  ^^j%  \]  o) 

wherein 

i  =  1 hj,        y  =  1 g^         1=1 7i^ 

The  multiples  of  ap_^_,_i,  namely  v,  a3,_:r+i,  where  Vt  is  prime  to  a^,  are 
divided  into  7i^  sets  of  g^  each ;  s^P  is  the  lowest  in  the  Z-th  set,  the  set  being 
4">  ^4"  •  •  •  •  »«''*~^4'')    andy  +  1  is  reduced  modulo^  g^.. 

489.  Theorem.  The  algebra  defined  by  the  groups  1  =  e^  =  e^  =  ej' 
e.^  ej  =  gj  63  eg  ej  =  fj  e^  e.,  e^  =  e^  63  e^  is  given  by  the  forms  "K  occurring 
in  the  equations 

where 

a;  =  p  +  l,^ 1,0    no  =  n      r2j,  +  i=l      7^,  i  =  1 "p-x+i     /=! «x 

/_,  is  any  integer  <  n^;  and  prime  to  n^.  [has  therefore  ^  (%)  values],  /+  1  is 
reduced  modulo  n^;  n^  is  any  divisor  of  w,  the  quotient^  being  ??p_^  +  i. 

490.  The  papers  of  Frobenius  and  Burnside  on  group-characteristics 
should  be  consulted. 


•Shaw  13. 


SPECIAL  CLASSES  OF  GROUPS  1  29 

XXVn.     SPECIAL  CLASSES  OF  GROUPS. 

491.  Since  every  group  determines  a  Fuoi{f:Niu.s  algebni,  it  is  evident  that 
this  algebra  might  be  used  to  determine  tlie  group  and  to  serve  in  applications 
of  the  group.  Since  the  group  admits  only  of  multiplication,  the  group 
properties  become  those  of  certain  numbers  in  the  algebra  combined  only  by 
multiplication.  However,  if  the  group  is  a  group  of  operators,  or  may  be 
viewed  as  a  group  of  operators,  it  may  happen  that  the  result  of  operating 
on  a  given  operand  may  be  additive,  in  which  case  the  numbers  of  the  algebra 
become  operators.     Examples  are  given  below. 

492.  Substitutions.  Since  every  abstract  group  of  order  r  is  isomorphic 
with  one  or  more  substitution  groups  on  r  letters  or  fewer,  it  follows  that  the 
permutations  or  substitutions  of  such  groups  may  be  expressed  by  numbers  of 
the  algebra  corresponding  to  the  abstract  group.  Thus  a  rational  integral 
algebraic  function  P  of  n  variables  may  be  reduced  to  the  form 

m 

P=^    Pi 

i=l 

where  Pj  is  expressible  in  the  form 

where  Af  is  a  positive  or  negative  numerical  coefficient  and  Sj  is  a  substitution 
of  the  symmetric  group  of  the  h  variables.  F^  is  a  rational  integral  algebraic 
function  of  the  variables.  All  the  substitutional  properties  of  P^  are  direct 
consequences  of  the  form  {A'p  + +  ^J;*  yS'„).     For  example 

-P  =  i  «3  —  i  «a  +  3af  CTj  —  3aj  ag  —  i  a|  Og  +  I  Oj  a5  =  Pi  4-  Pg 
where 

P]=  i  [1  —  (rto  a-i)  +  («i  Ws)  —  («1  «2  «3)]  •  «3 

Pi=     [3  —  s  («i  a.  Os)  —  3  (rt,  ttg)  +  I  {(ii  as)]  .  a;  a, 

wherein  the  bracket  expresses  an  operation.  We  may  find  solutions  for 
equations  such  as 

{l+a  +  a^  +  a')P=0  <t  =  (abed) 

or  other  forms  in  which  the  parenthesis  is  any  rational  integral  expression  in 
terms  of  substitutions. 

The  solution  of  this  particular  case  is  P  =  (l  —  a) F,  where  F  is  any 
rational  integral  function.  These  equations  are  useful  in  the  study  of 
invariants.' 

493.  Linear  Groups.  A  group  of  linear  substitutions  has  corresponding  to 
it  an  abstract  group,  such  that  if  the  generating  substitutions  of  the  linear 

group,  H,  are  2i,  ^2 2p,  with  certain  relations  2i,  2^, -r,  =  1,    2,-, 2^ 

....  2,.,  =  1,  etc.,  then  the  abstract  group  is  determined  by  generating  sub- 
stitutions (Ti,  (Tj Op,  with  relations  (7^,0^, a,,  =  1,   aj, cr^, <^r,  =  1,  etc. 


'A.  TocNO  1. 


130  SYNOPSIS  OF  LINEAK  ASSOCIATIVE  ALGEBRA 

If  we  choose  a  suitable  polygon  in  a  fundamental  circle,  the  circle  is 
divisible  into  an  infinity  of  triangles,  which  may  be  produced  by  inversions  at 
the  corners  of  the  polygon,  according  to  the  well-known  methods.  The 
group  G  generated  by  (Tj,  cTg.  .  .  -cTp  without  the  relations  is  in  general  infinite. 
With  the  addition  of  the  relations  we  get  a  group  G'  isomorphic  with 
E,  H  being  merihedrically  isomorphic  with    G. 

Then  G',  or  what  is  the  same  thing  H,  may  be  made  isomorphic  with  a 
Frobenitjs  algebra,  which  is  of  use  in  the  applications  of  the  group.  A  notable 
application  of  this  kind  was  made  by  Poincare.^ 

This  application  is  devoted  on  the  one  hand  to  the  study  of  the  linear 
groups  of  the  periods  of  the  two  kinds  of  integrals  of  a  linear  differential 
equation  of  order  n  which  is  algebraically  integrable;  and,  on  the  other,  to  the 
proof  that  for  every  finite  group  contained  in  the  general  linear  group  of  n 
variables  there  is  such  a  differential  equation.  The  results  are  chiefly  the 
following : 

494.  Theorem.  For  every  group  G'  there  is  a  system  of  Fuchsian  func- 
tions, Abelian  integrals  of  the  first  kind,  such  that  if  ^(2)  is  any  such  function, 
and  if  S  is  any  substitution  of  G'  to  which  corresponds  a  linear  substitution 

on^,  "^f,    (a^-^X=l),    then 

where  co  is  called  a  i^eriod  of  K(z). 

There  are  also  Abelian  integrals  of  the  second  kind  P{z,  a),  such  that 
P{zS,  a)  ■=.  P(z,  a)  -\-  ^{a),  where  ^(a)  is  the  a-derivative  of  a  function  of  the 
first  kind. 

495.  Theorem.  The  genus  of  the  group  being  q,  there  are  q  independent 
integrals  of  each  kind ;  all  others  are  expressible  linearly  in  terms  of  these. 

496.  Theorem.  In  the  Frobenius  algebra  corresponding  to  the  group  let 
X  be  2  .  -^iCi,  where  e,-  corresponds  to  S^.  Then  KX  means  I.XiK{za~^),  and 
c)X  means  the  period  of  KX  corresponding  to  the  period  u  of  Kz.  Then  there 
are  three  kinds  of  quadrates  in  the  algebra. 

I.  Those  for  which  KX^  =  constant,  for  all  values  of  K  and  any  number 
Xa  in  the  a-th  quadrate.  In  this  case  (dX=  0  identically  for  any  ^and  any 
substitution  S,  and  if  A'„  =  2X„;<'j,  there  are  linear  relations  among  the 
coefficients  X^i.     Also  P{z,  a)  X,  is  an  algebraic  function. 

II.  Those  for  which  KX^  is  constant  for  each  K  if  X,  is  properly  chosen, 
BO  that  for  any  ^and  any  S  there  is  an  X^  such  that  uX^  =  0. 

There  is  an  integral  K'  whose  periods  are  linear  combinations  with 
integral  coefficients  of  X^t ;   this  integral  K'  combined  with  K  by  Riemann's 


'  POINCAR^  4. 


SPECIAL  CLASSES  OF  GROUPS  1  31 

relations  gives  tlie  coefficients  of  the  periods  o  in    uX  =  ^  JC^iUf-,   that  is, 
determines  uf^j.     There  are  but  q  such  relations  independent. 

Also  P{z,  a)  X^  is  not  an  algebraic  function  for  this  X^. 

III.  K .  X^  is  not  constant  for  certain  /(T's,  and  any  X^.  For  any  such 
K  we  may  write  KX^  =  (^(z),  then  the  periods  of  G{z)  being  cji,  cjg  •  •  •  •  <-<,„ 
for  the  m  substitutions  of  G\  if  we  forin  the  periods  of  G{za),  we  get  the 
same  periods  w  in  another  order  ;  a  group  determinant  may  be  formed  from 
these  by  letting  a  run  through  G,  which  must  vanish  as  well  as  its  minors  of 
the  first  in  —  n  —  1  orders. 

That  there  be  a  rational  function  of  x,  y,  satisfying  a  linear  equation  of 
order  n,  it  is  necessary  and  sufficient  that  there  are  numbers  h^,  h^  . .  ■  ■  h^ 
whose  group  determinant  is  of  the  character  above.  There  is  thus  always  at 
least  one  quadrate  of  the  third  kind. 

497.  Theorem.  An  integral  of  the  first  kind,  K,  helongs  to  a  quadrate  if 
^X  =  constant,  for  any  number  X  not  in  this  quadrate,  but  KX  is  not  con- 
stant for  all  numbers  in  the  quadrate. 

An  integral  P{z,  a)  helongs  to  a  quadrate  if  for  all  values  of  X  not  in  this 
quadrate  P{z,  a)  X  is  an  algebraic  function  ;  but  for  some  values  of  X  in  the 
quadrate  P{Zy  a)  X  is  not  an  algebraic  function. 

The  number  of  integrals  of  the  first  kind  belonging  to  a  quadrate  of 
order  a?  is  a  multiple  of  a. 

Any  integral  can  be  separated  into  integrals  each  of  which  belongs  to  a 
single  quadrate. 

498.  Theorem.  The  2q  periods  of  K{z)  are  subject  to  a  linear  transforma- 
tion by  each  substitution  S  of  G'.  The  totality  of  these  linear  transformations 
furnish  a  linear  group  isomorphic  with  H. 

The  relations  between  the  periods  of  P{z,  a)  are  found  by  writing  the 
linear  relations  between  K{z),  K{zS^,  K{zS.?i,  etc.,  and  differentiating  them. 
The  derivatives  are  subject  to  linear  transformations  which  also  generate  a 
group  isomorphic  with  H. 

The  second  group  is  related  to  the  totality  of  quadrates  of  the  second 
kind,  the  first  group  to  the  totality  of  quadrates  of  the  third  kind. 

499.  Modular  Group.  This  has  been  studied  by  means  of  the  commutative 
algebras.^ 

500.  Laurent'  has  made  use  of  representations  of  linear  substitutions  by 
quadrate  numbers  or  tettarions,  to  derive  several  theorems.  His  processes  are 
briefly  indicated  below. 

Theorem.   If  cr  =  2  c^j  \j,  where 

c„  =  1  ?■  =  1 11  Cij  =  —Cji  i:^J 

then  the  tettarion  r  =  2cr^  —  1  represents  an  oWAogrona?  substitution,  and  the 

'  J.  W.  Young  1.  'Laukent  1,  2,  3,  4. 


132  SYNOPSIS  OF  LINEAK  ASSOCIATIVE  ALGEBRA 

orthogonal  group  consists  of  all  such  substitutions.  In  this  case  a  represents 
a  sJceiv  substitution. 

501.  Theorem.  Every  orthogonal  substitution  may  be  represented  by  the 
product  of  tettarions  of  the  type  ^ 

w  =  /ln  +  ^22+  ••••  +Kn  +  i^ii  +  hj)  cos  ^  +  i^Hj  —  T^ji)  sin  <?> 
Zji  and  2,jj  absent 

502.  Theorem.  Tettarions  of  the  type  c  and  1  +  c  (/l,^  +  ^^i)  produce 
tettarions  representing  symmetric  substitutions. 

503.  Theorem.  Tettarions  of  the  type  1  +  /l,j  produce  tettarions  which 
represent  substitutions  with  integral  coeflRcients. 

l....n 

504.  Theorem.    If  Tr=:2  a^^  Aj^  represents  an  orthogonal  substitution,  then 

l....n 

Tpj  =  2  aip  ttjg  /ii^   gives    a  new  group  of  linear  substitutions.     By  similar 

compounding  of  coefficients  of  known  groups,  new  groups  may  be  formed. 

505.  AuTONNE^  has  applied  the  theory  of  matrices  to  derive  theorems 
relating  to  linear  groups,  real,  orthogonal,  hermitian,  and  hypohermitian.     If 

where  Uij  is  the  conjugate  of  the  ordinary  complex  number  a^j,  then  t  is 
symmetric  if  ?  =  t  ;  it  is  orthogonal  if  tt  =  1  ;  real  if  t=:t;  unitary  if 
TT=  1 ;  hermitian  if  t=t.  In  the  latter  case  the  hermitian  form  /.  ^  (t)  ^  >•  0. 
[In  this  expression  (r)  acts  on  ^  as  a  linear  vector  operator].  If  r  is  hermitian 
there  is  one  and  only  one  hermitian  ^  such  that  ^"  =  r,  or  <f  =  t*. 

Theorem.  That  an  n-ary  group  G  can  be  rendered  real  and  orthogonal 
by  a  convenient  choice  of  variables,  the  following  conditions  are  necessary 
and  sufficient : 

(1)  G  possesses  two  absolute  invariants:  a  hermitian  form  I .  ^(r)^  and 
an  n-ary  quadratic  form  of  determinant  unity,  P  =■  I .  ^(p)^. 

(2)  G  having  been  rendered  unitary  by  being  put  into  the  form  t*  Gt~^, 
in  the  transform  of  P,  p  is  unitary. 

506.  Theorem.  Every  tettarion  is  the  product  of  a  unitary  tettarion  by  a 
hermitian  tettarion. 

To  put  a  into  such  a  form  we  take  t"  =  aa  and  d  =  af^;    then  a  =:  vt. 
The  literature   of  bilinear  forms    furnishes    many    investigations    along 
these  lines. 

I  Cf.  Taber  C,  7,  and  other  papers  on  matrices.  ■  Autonne  1,  2,  3,  4. 


MODULAR  SYSTEMS  133 

XXVm.     DrPFERENTIAL  EQUATIONS. 

507.  Pfajf's  Equation.   To  the  Bolution  of  the  equation 

Xj  dxi  +  X^  dx.,  -f-  .  .  .  .  -f-  Xm  dxm  =  0 
Grassmann'  applied  the  methods  of  the  Ausdahnungslehre. 

508.  La  Place's  Equation.  This  may  be  written  y^y— -q.  ft  has  been 
treated  by  quaternionic  methods  in  the  case  of  three  variables.^  Other  equa- 
tions and  systems  of  equations  which  appear  in  physics  have  been  handled  in 
analogous  ways.  The  literature  of  quaternions  and  vector  analysis  should  be 
consulted.^  The  full  advantage  of  treating  the  general  operator  v  ^s  an 
associative  number,  would  .simplify  many  problems  and  suggest  solutions  for 
cases  not  yet  handled. 

509.  It  is  pointed  out  by  Skill*  that  by  means  of  matrices  the  operator 

g2  02  32  ^      82  3'  7       3' 

^  =  ''  a.r^  +  *  3/  +  ^   dz"-  +  f  dydz  +  ^  ~dzdx  +  ^  Tx 37 

can  be  factored  into 

{«!,  +  (^-«i>)  a^  +  {9-n)^\     { s-^  +  i' 37  +  ? -3z  } 

p  and  q  being  matrices  (tettarions). 

Therefore  any  matrical  function  of  x,  y,  z  which  vanishes  under  the 
operation  of  either  of  these  linear  operators  is  a  solution  of  the  equation 

A  .  0=0 

It  is  obvious  that  this  method  is  capable  of  considerable  extension.* 


XXIX.     MODULAR  SYSTEMS. 

510.  It  is  obvious  that  every  multiplication  table  may  be  expressed  in 
the  form 

gje^— 2yyfce*=0 

If  now  we  consider  a  domain  admitting  e,-,  e^,  etc.,  and  their  products  and 
linear  combinations,  it  is  evident  that  we  have  a  modular  system.  The 
expressions  e^  .  .  . .  need  not  be  ordinary  algebraic  variables,  of  course ;  they 
may  be  function-signs,  for  example. 

Every  modular  system  may  be  considered  to  represent,  and  may  be 
represented  by,  an  algebra.  From  this  point  of  view  all  numbers  are  quali- 
tative except  integers. 


>FoR8TTHE  1.     Cf.  Ausdehnungslehre,  1862,  §§500-527. 
'  Boole  1;    Carmichael  1,  2,  3,  4;    Brill  2 ;   Graves  1. 

»  WeDDERBCRN  3  ;     POCKLINOTON  1.  *BrILL3. 

•  Cf.  B.  Peikoe  2.     Same  In  Appendix  I  In  B.  PeirceS. 

9 


134  SYNOPSIS  OF  LINEAR  ASSOCIATIVA  ALGEBRA 

XXX.     OPERATORS. 

511.  The  use  of  different  abstract  algebras  in  forms  which  practically 
make  them  operators  on  other  entities  is  quite  common  in  some  directions. 
In  such  applications  the  theory  demands  a  consideration  of  the  operands  as 
much  as  of  the  operators.  As  operators  they  have  also  certain  invariant, 
covariant,  contravariant,  etc.  operands,  so  that  the  invariant  theory  becomes 
important. 

For  example,  the  algebra  of  nonions  plays  a  very  important  part  in 
quaternions  as  the  theory  of  the  linear  vector  operator.^ 

512.  Invariants  of  Qnantics.  The  formulae  and  methods  of  quaternions 
have  been  applied  to  the  study  of  the  invariants  of  the  orthogonal  trans- 
formations of  ternary  and  quaternary  quantics.^  If  ^  is  a  vector,  then  q^g~^ 
is  an  orthogonal  transformation  of  ^,  q  being  any  quaternion  of  non-vanishing 
tensor.  Every  vector  or  power  of  a  vector  or  products  of  powers  of  vectors 
furnishes  a  pseudo-invariant.  Orthogonal  ternary  invariants  are  then  those 
functions  of  vectors  which  are  mere  scalars,  the  list  being  as  follows : 

r-p      r-a     Sa^     Spa^     Sa^y 

In  these,  a,  /?,  etc.  are  practically  different  nablas  operating  on  p,  so  that 
we  understand  by  S  .  a^y  substantially  what  is  also  written  aS.ViVsVs- 
The  formulae  of  quaternions  become  thus  applicable  to  these  symbolic 
operators,  yielding  reductions,  syzygies,  etc.     For  example,  the  syzygies 

Sa^  Syhe  —  S^y  Sahe  +  Sl^h  Says  —  S(5e  So.yh  =  0 
Sap  S^yh  -  Sl5p  Sayh  +  Syp  Sai^h  -  Shp  Sa^y  =  0 

This  amounts,  of  course,  to  a  new  interpretation  of  Aronhold's  notation,  and 
the  process  may  readily  be  generalized  to  n  dimensions  by  introducing  the 
forms    7a/3,  Ipp,  lap,  and  the  like. 

513.  Differential  Operators.  The  differential  operators  occurring  in  con- 
tinuous group-theory  are  associative,  hence  generate  an  associative  algebra 
(usually  infinite  in  dimensions).  Groups  of  such  operators  are  groups  in  the 
algebras  they  define,  and  their  theory  may  be  considered  to  be  a  chapter  on 
group-theory  of  infinite  algebras.  The  whole  subject  of  infinite  algebras  is 
undeveloped.  The  iterative  calculus,  the  calculus  of  functional  equations, 
and  the  calculus  of  linear  operations  are  closely  connected  with  the  subject 
of  this  memoir.^ 

■  See  references  under  NonionB,  previously  given. 

»  McMahon  1 ;   Shaw  14. 

'PiNOHERLE  2,  3;  Lemekat  1,  2,  3;   Leac  1.     The  literature  of  this  subject  should  be  consulted. 


BIBLIOGEAPHY 


Some  titles  appear  in  this  list  because  they  bear  on  the  subject,  though  they  are  not 
referred  to  in  the  paper.  Those  referred  to  in  the  paper  are  numbered  as  in  the  reference. 
The  list  is  compiled  partly  from  the  "Bifjliography  of  Quaternions,  etc.,"  partly  from  a 
"  Hibliogniphy  of  Matrices"  in  manuscript  by  Mr.  J.  II.  Maclagan-Wedderburn,  partly 
from  the  author's  notes.  Additional  titles  are  solicited.  The  bibliographies  mentioned 
should  be  consulted  for  further  titles. 

The  number  after  the  year  is  the  volume;  or,  if  in  parentheses,  the  series  followed 
by  the  volume.     Succeeding  numbers  are  pages. 

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31;  268-271. 

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136-141. 

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delle  equipoUenze.      Ven.  1st.  Mem.  (1848)  1;  235-267. 

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e  Ridolfl.      Ven.  1st.  Atti  (1847)  (1),  6;  53-59. 

8.  Saggio  auir  algebra  degli  immaginarie.      Ven.  1st.  Mem.  (1852)  4;  243-344. 

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Atti  (1858)  (3),  1 :  334-343.     Alao  Mem.  Soc.  Hal.  (1863)  1;  126-186. 

11.  Spoaizione  del  nuovi  metodl  di  geometria  analitica.      Vni.  1st.  Mem.  (1860). 

12.  Sul  calcolo  dei  quatemioni.     Uudecima  Eiviata  di  Giomali.      Ven.  1st.  Atti  (1871)  (2),  204. 

13.  Sul  calcolo  dei  quatemioni  osaia  teoria  dei  rapporti  geometrici  nello  spazio.     Daodecima  Rivista  di 

Giomali.      Ven.  1st.  Atti  (1873)  (3),  69. 

14.  Exposition  de  la  methode  des  ^quipollencea  de  Giuato  Bellavitis;    traduction  par  C.  A.  Laisant. 

Nouv.  Ann.  (1873-74)  (2),   12,  13. 

15.  SuUe  origini  del  metodo  delle  equipoUenze.      Ven.  Itt.  Mem.  (1876)  19;  449-491. 

16.  Sur  la  these  de  M.  Lalaant  relative  au  calcul  dea  quaternions.     Qnatuordicesima  Riviata  di  Giornali 

Ven.  1st.  Atti  (1878)  (2),  116. 

136 


136  SYNOPSIS  OP  LINEAR  ASSOCIATIVE  ALGE3BRA 

Beman,  Wooster  Woodrnff. 

1.  A  brief  account   of  the  essential  features   of  Grassmann's   extensive   algebra.     Analyst  (188!)  8; 

96-97,  114-134. 

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Beklott,  B. 

1.   Th^orie  des  qnantlt^s  complexes  a  «  unites  priucipales.     (1886.)     Thdse.     Paris  (Gauthier-Vlllars), 
126  pp. 

BoCHEB,  Maxlme. 

The  geometric  representation  of  iraaginaries.     Annals  of  Math.  (1893)7;  70-72. 
1.    Conceptions  and  methods  of  mathematics.     Amer.  Math.  Soc.  Bull.  (1904)  (3),  11;   136-135. 

Boole,  George. 

I.  Application   of  the  method  of  quaternions  to   the  solution   of   the   partial  differential  equation 

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Beili,,  John. 

1.  A   new  geometrical   representation   of   the  quaternion   analysis.      Proc.   Camb.  Phil.  Soc.  (1889)  6, 

151-1.59. 

2.  Note  on  the  application  of  quaternions  to  the  discussion  of  Laplace's  equation.     Proc.  Camb.  Phil. 

Soc.  (1892)  7;   130-135,  151-156. 

3.  On  the  application  of  the  theory  of  matrices  to  the  discussion  of  linear  differential  equations  with 

constant  coefficients.     Proc.  Camb.  Phil.  Soc.  (1894)  8  ;  301-310. 

BtTCHEKEK,  A.  H. 

1.    Elemente  der  Vectoranalyse.     Leipzig  (Teubner).     (1903)  6  +  91  pp. 

Btjchhbim,  Arthur. 

1.    On  the  application  of  quaternions  to  the  theory  of  the  linear  complex  and  the  linear  congruence. 

Messenger  of  Math.  (1883)  (3),  18;   139-130. 
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3.  On  the  theory  of  matrices.     Lond.  Math.  Soc.  Proc.  (1884)  16;   63-83. 

4.  On  the  theory  of  screws  in  elliptic  space.     Lond.  Math.  Soc.  Proc.   (1884-87)  15;   83-98:  16;   15-37: 

17;  240-S.54:  18;  88-96. 

5.  A  memoir  on  biquaternions.     Amer.  Jour,  of  Math.  (1885)  7;   293-336. 

6.  On  Clifford's  theory  of  graphs.     Lond.  Math.  Soc.  Proc.  (1886)  17;  80-106. 

7.  An  extension  of  a  theorem   of   Prof.   Sylvester's   relating  to   matrices.     Phil.  Mag.  (1886)   (5),  22; 

173-174. 

8.  Note  on  double  algebra.     Messenger  of  Math.  (Wfi7)  16;  62-63. 
Note  on  triple  algebra.     Messenger  of  Math.  (1887)  16;   111-114. 

9.  Proof  of  Sylvester's  "Third  Law  of  Motion."     Phil.  Mag.  (1884)  (.5),  18;  4.59-4B0. 

10.    On  a  theorem  of  Prof.  Klein's  relative  to  symmetric  matrices.     Messenger  of  Math.  (1887)  17;  79. 

II.  Note  on  matrices  in  involution.     Messenger  of  Math.  (1S89)  18;  103-104. 

Bcknside,  William. 

1.  On  the  continnous  groups  defined  by  finite  groups.    Lond.Math.Soc.Proc.  (1898)  39;  307-324,  546-.565. 

2.  On  group  characteristics.     Lond.  Math.  Soc.  Proc.  (1901)  SS;  li6-162. 

3.  On  some  properties  of  groups  of  odd  order.      Lond.  Math.  Soc.  Proc.  (1901)  33;   163-185. 

4.  On  linear  substitutions,  etc.     Messenger  of  Math.  (1904)  38  ;  133. 

5.  On  the  representation  of  a  group  of  finite  order  as  an  irreducible  group  of  linear  substitutions  and 

the  direct  establishment  of  the  relations  between  the  group  characteristics.    Lond.  Math.  Soc.  Proc. 
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Carstens,  R.  L. 

1.   Definition  of  quaternions  by  Independent  postulates.     Amer.  Math.  Soc.  Bull.  (1906)  (3),  13  ;  392-394. 
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Sur  la  Btrncturo  des  gronpes  de  transformations  finis  et  continns.     These.     Paris  (Gauthier-Villars) 

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138  SYNOPSIS  OF  LINEAR  ASSOCIATIVE  ALGEBRA 

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140  SYNOPSIS  OF  LINEAR  ASSOCIATIVE  ALGEBRA 

Jaenee,  E. 

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142  SYNOPSIS  OF  LINEAK  ASSOCIATIVE  ALGEBRA 

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Matrices  which  represent  vectors.     Boston  Tech.  Quar.  (1893)  6;  348-351. 

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